QH105 


STATISTICAL  METHODS 


WITH  SPECIAL    REFERENCE  TO 


BIOLOGICAL  VARIATION. 


BY 

C.  B.  DAVENPORT, 

Director  of  Department  of  Experimental  Evolution, 
Carnegie  Institution  of  Washington. 


THIRD,  REVISED  EDITION. 
FIRST   THOUSaJSTI'.  • 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 
LONDON:  CHAPMAN   &  HALL,   LIMITED 


Copyright,  1899,  1904, 

BY 

C.  B.  DAVENPORT. 


PRESS  OF 

BRAUNWORTH    &   CO. 

BOOK   MANUFACTURERS 

BROOKLYN,    N.   Y. 


PKEFACE. 


THIS  book  has  been  issued  iu  answer  to  a  repeated  call  for  a 
simple  presentation  of  the  newer  statistical  methods  in  their 
application  to  biology.  The  immediate  need  which  has  called 
it  forth  is  that  of  a  handbook  containing  the  working  formulae 
for  use  at  summer  laboratories  where  material  for  variation - 
study  abounds.  In  order  that  the  book  should  not  be  too 
bulky  the  text  has  been  condensed  as  much  as  is  consistent 
with  clearness. 

This  book  was  already  in  rough  draft  when  the  work  of 
Duncker  appeared  in  Roux's  Archiv.  I  have  made  much  use 
of  Duncker's  paper,  especially  in  Chapter  IV.  I  am  indebted 
to  Dr.  Frederick  H.  Safford,  Assistant  Professor  of  Mathe- 
matics at  the  University  of  Cincinnati  and  formerly  Instructor 
at  Harvard  University,  for  kindly  reading  the  proofs  and  for 
valuable  advice.  To  Messrs.  Keuffel  and  Esser,  of  New  York, 
I  am  indebted  for  the  use  of  the  electrotypes  of  Figures  1  and  2. 
Finally,  I  cannot  fail  to  acknowledge  the  cordial  cooperation 
which  the  publishers  have  given  in  making  the  book  ser- 
viceable. 

C.  B.  DAVENPORT. 

BIOLOGICAL  LABORATORY  OF  THE  BROOKLYN  INSTITUTE, 

COLD  SPRING  HARBOR,  LONG  ISLAND, 

June  29, 1899. 

iii 


383952 


PREFACE  TO  THE  SECOND  EDITION. 


THE  first  edition  of  this  book  having  been  favorably  re- 
ceived, the  'publishers  have  authorized  a  revised  edition 
embodying  many  of  the  new  statistical  methods  elaborated 
chiefly  by  Professor  Karl  Pearson  and  his  students  and 
associates,  and  presenting  a  summary  of  the  results  gained 
by  these  methods.  These,  while  increasing  somewhat  the 
bulk  of  the  book,  have,  it  is  hoped,  rendered  it  more  service- 
able to  investigators.  Too  much  emphasis  can  hardly  be 
laid  on  the  debt  that  Biometriciajis  owe  to  Professor  Pear- 
son's indefatigable  researches  in  the  new  science  of  Biome- 
try— especially  in  the  development  of  Statistical  Theory. 

The  publishers,  also,  of  this  book  are  deserving  of  credit 
for  the  courage  they  have  shown  in  reproducing  expensive 
tables  for  the  use  of  a  still  very  limited  body  of  statistical 
workers.  Especial  attention  is  called  to  Table  IV,  which 
is  an  extension  of  Table  IV  of  the  first  edition  that  was  cal- 
culated by  Professor  Frederick  H.  Safford,  and  appears  to 
have  been  the  first  published  table  of  the  normal  probability 
integrals  based  on  the  standard  deviation.  More  recently 
Mr.  W.  F.  Sheppard  has  published  in  Biometrika  a  similar 
table  in  which,  however,  the  tabular  entries  are  given  to 
seven  places  of  decimals,  while  the  arguments  are  given 
to  two  decimal  places  only.  In  the  present  table  the  argu- 
ments are  subdivided  to  three  places  of  decimals  and  with 
the  aid  of  the  table  of  proportional  parts  interpolation  is 
easily  effected. 

Especial  acknowledgment  must  be  made  of  assistance 
received  from  my  friend  Mr.  F.  E.  Lutz,  who  read  over  the 
entire  manuscript  and  contributed  certain  of  the  numerical 
examples. 

STATION  FOR  EXPERIMENTAL  EVOLUTION 

CARNEGIE  INSTITUTION  OF  WASHINGTON. 

COLD  SPRING  HARBOR, 

March  27,  1904. 

iv 


CONTENTS. 


CHAPTER  I. 

ON  METHODS  OF  MEASURING  ORGANISMS. 

PAGE 

Preliminary  definitions 1 

Methods  of  collecting  individuals  for  measurement 2 

Processes  preliminary  to  measuring  characters 2 

The  determination  of  integral  variates — -Methods  of  counting 3 

The  determination  of  graduated  variates — Method  of  measurement .  4 

Straight  lines  on  a  plane  surface 4 

Distances  through  solid  bodies  or  cavities 4 

Area  of  plane  surfaces 4 

Area  of  a  curved  surface 5 

Characters  occupying  three  dimensions  of  space 6 

Characters  having  weight 6 

Color  characters 6 

Marking-characters 7 

Aids  in  calculating 7 

Precautions  in  arithmetical  work 8 

CHAPTER  II. 

ON  THE  SERIATION  AND  PLOTTING  OF  DATA  AND  THE  FREQUENCY 
POLYGON. 

Seriation 10 

Plotting 11 

Method  of  rectangles 11 

Method  of  loaded  ordinates 12 

The  rejection  of  extreme  variates 12 

Certain  constants  of  the  frequency  polygon 13 

The  average  or  mean 13 

The  mode 13 

The  median  magnitude 14 

The  probable  error  of  the  determination 14 

The  probable  difference  between  two  averages 15 

The  probable  error  of  the  mean 15 

The  probable  error  of  the  median 15 

The  geometric  mean 15 

The  index  of  variability 15 

The  probable  error  of  the  standard  deviation 16 

Average  deviation  and  probable  departure 16 

Y 


VI  CONTENTS. 

PAGE 

Coefficient  of  variability 16 

The  probable  error  of  the  coefficient  of  variability 16 

Quick  methods  of  roughly  determining  average  and  variability. ...      17 

CHAPTER  III. 

THE  CLASSES  OF  FREQUENCY  POLYGONS. 

Classificatfon 19 

To  classify  a  simple  frequency  polygon 19 

The  normal  curve 22 

To  compare  any  observed  curve  with  the  theoretical  normal 

curve 23 

The  index  of  abmodality 23 

To  determine  the  closeness  of  fit  of  a  theoretical  polygon  to  the 

observed  polygon 24 

To  determine  the  probability  of  a  given  distribution  being 

normal 24 

The  probable  range  of  abscissae 25 

The  normal  curve  as  a  binomial  curve 25 

Example  of  a  normal  curve 26 

To  find  the  average  difference  between  the  pth  and  the  (p+l)th 

individual  in  any  seriation 27 

To  find  the  best  fitting  normal  frequency  distribution  when  only  a 

portion  of  an  empirical  distribution  is  given 28 

Other  unimodal  frequency  polygons 30 

The  range  of  the  curve 30 

Asymmetry  or  skewness 30 

To  compare  any  observed  frequency  polyg-on  of  Type  I  with  its 

corresponding  theoretical  curve 31 

To  compare  any  observed  frequency  polygon  of  Type  II  with 

its  corresponding  theoretical  curve 32 

To  compare  any  observed  frequency  polygon  of  Type  III  with 

its  corresponding  theoretical  curve 33 

To  compare  any  observed  frequency  curve  of  Type  IV  with  its 

corresponding  theoretical  curve 33 

To  compare  any  observed  frequency  polygon  of  Type  V  with 

its  corresponding  theoretical  curve 34 

To  compare  any  observed  frequency  polygon  of  Type  VI  with 

its  corresponding  theoretical  curve 34 

Example  of  calculating  the  theoretical  curve  corresponding  with 

observed  data 35 

The  use  of  logarithms  in  curve  fitting 36 

General    38 

Type  IV.  ., 

Multimodal  curves 39 

CHAPTER  IV. 

CORRELATED  VARIABILITY. 

General  principles 42 

Methods  of  determining  coefficient  of  correlation 44 


CONTENTS.  Til 


PAGE 

Galton's  graphic  method 44 

Pearson's  method 44 

Brief  method 45 

Probable  error  of  r 45 

Example 45 

Coefficient  of  regression 47 

The  quantitative  treatment  of  characters  not  quantitatively  meas- 
urable   47 

The  correlation  of  non-quantitative  qualities 49 

Example. 51 

Quick  methods  of  roughly  determining  the  coefficient  of  correlation.  54 

Spurious  correlation  in  indices 54 

Heredity 55 

Uniparental  inheritance 55 

Biparental  inheritance 55 

To  find  the  coefficient  of  correlation  between  brethren  from  the 

means  of  the  arrays 56 

Galton's  law  of  ancestral  heredity 57 

Mendel's  law  of  inheritance  in  hybrids 57 

A  dissymmetry  index 60 

CHAPTER  V. 
SOME  RESULTS  OF  STATISTICAL  BIOLOGICAL  STUDY. 

General 62 

Variability 62 

General 62 

Man , 63 

Mammalia 65 

Aves 65 

Amphibia 66 

Pisces 66 

Tracheata 66 

Crustacea 66 

Annelida 67 

Brachiopoda 67 

Bryozoa 67 

Mollusca 67 

Echinodermata 68 

Ccelenterata 68 

Protista 69 

Plants 69 

Some  types  of  biological  distributions.  . 71 

Type  1 71 

Type  IV 72 

Type  V 72 

Normal 72 

Skewness 72 

Complex  distributions 73 


Vlll  COKTEOTS. 


PAGE 

Correlation 73 

General 73 

Man 73 

Lower  animals 76 

Plants 78 

Heredity 78 

General 78 

Parental 79 

Grandparental -80 

Fraternal 80 

Theoretical  coefficient  of  heredity  between,  relatives 81 

Homotyposis 81 

Mendelism 82 

Telegony 82 

Fertility 82 

Selection 82 

Dissymmetry 82 

Direct  effect  of  environment 83 

Local  races 83 

Useful  tables 84 

BIBLIOGRAPHY 85 

EXPLANATION  OF  TABLES 105 

LIST  OF  TABLES. 

The  Greek  alphabet 114 

Index  to  the  principal  letters  used  in  the  formulse  of  this  book. .  .  115 

Table         I.  Formulas 116 

II.  Certain  constants  and  their  logarithms 117 

*'         III.  Table  of  ordinates  of  normal  curve,  or  values  of  — 

2/o 

corresponding  to  values  of  — 118 

a 

•'          IV.  Table  of  half-class  index  values   (£a)   or  the  values 
of  the  normal  probability  integral  corresponding  to 

values  of  — ;  or  the  fraction  of  the  area  of  the  curve 
a 

between  the  limits  0  and  -\ — -  or  0  and  — - 119 

a  -a 

"           V.  Table  of  Log  r  functions  of  p 126 

"          VI.  Tat>le  of  reduction  of  linear  dimensions,  from  common 

to  metric  system 128 

"        VII.  Minutes  and  seconds  in  decimals  of  a  degree 128 

•'      VIII.  First  to  sixth  powers  of  integers  from  1  to  50 129 

*'          IX.  Probable  errors  of  the  coefficient  of  correlation 130 

•'           X.  Squares,  cubes,  square-roots,  cube-roots,  and  recip- 
rocals   131 

**          XI.  Logarithms  of  numbers 149 

••        XII.  Logarithmic  sines,  cosines,  tangents,  and  cotangents.  176 


STATISTICAL  METHODS 

WITH    SPECIAL    REFERENCE   TO 

BIOLOGICAL  VAKIATION. 


CHAPTER  I. 

ON  METHODS  OF  MEASURING  ORGANISMS.  . 
Preliminary  Definitions. 

An  individual  is  a  segregated  mass  of  living  matter,  capable 
of  independent  existence.  Individuals  are  either  simple  or 
compound,  i.e.,  stocks  or  corms.  In  the  case  of  a  compound 
individual  the  morphological  unit  may  be  called  a  person. 

A  multiple  organ  is  one  that  is  repeated  many  times  on  the 
same  individual.  Example,  the  leaves  on  a  tree,  the  scales 
on  a  fish. 

A  character  is  any  quality  common  to  a  number  of  indi- 
viduals or  to  a  number  of  multiple  organs  of  one  individual. 

A  variate  is  a  single  magnitude-determination  of  a  character. 

Integral  variates  are  magnitude-determinations  of  charac- 
ters which  from  their  nature  are  expressed  in  integers.  Such 
magnitudes  are  expressed  by  counting;  e.g.,  the  number  of 
teeth  in  the  porpoise.  These  are  also  called  discontinuous. 

Graduated  variates  are  magnitude-determinations  of  char- 
acters which  do  not  exist  as  integers  and  which  may  conse- 
quently differ  in  different  variates  by  any  degree  of  magni- 
tude however  small;  e.g.,  the  stature  of  man. 

A  variant,  among  integral  variates,  is  a  single  number-con- 
dition, e.g.,  5  (flowers),  13  (ray-flowers),  etc. 

A  class,  among  graduated  variates,  includes  variates  of 
the  same  or  nearly  the  same  magnitude.  The  class  range 
gives  the  limits  between  which  the  variates  of  any  class  fall. 

Individual  variation  deals  with  diversity  in  the  characters 
of  individuals. 

Organ  variation,  or  partial  variation,  deals  with  diversity  in 
multiple  organs  in  single  individuals. 

1 


2  ,    STATISTICAL   METHODS. 

Met5:orto  of  Collecting  Individuals  for  Meas- 
urement, 

In  collecting  a  lot  of  individuals  for  the  study  of  the  varia- 
bility of  any  character  undue  selection  must  be  avoided.  The 
rule  is: 

Having  settled  upon  the  general  conditions,  of  race,  sex, 
locality,  age,  which  the  individuals  to  be  measured  must  fulfil, 
take  the  individuals  methodically  at  random  and  without  possible 
selection  of  individuals  on  the  basis  of  the  magnitude  of  tJie 
character  to  be  measured.  If  the  individuals  are  simply  not 
consciously  selected  on  the  basis  of  magnitude  of  the  character 
they  will  often  be  taken  sufficiently  at  random. 

The  number  ot  variates  to  be  obtained  should  be  large;  if 
possible  from  200  to  2000,  depending  on  abundance  and 
variability  of  the  material. 

Processes  Preliminary  to  Measuring 
Characters. 

Some  characters  can  best  be  measured  directly;  e.g.,  the 
stature  of  a  race  of  men.  Often  the  character  can  be  better 
studied  by  reproducing  it  on  paper.  The  two  principal 
methods  of  reproducing  are  by  photography  and  by  camera 
drawings. 

For  photograpliic  reproductions  the  organs  to  be  measured 
will  be  differently  treated  according  as  they  are  opaque  or 
transparent.  Opaque  organs  should  be  arranged  if  possible 
in  large  series  on  a  suitable  opaque  or  transparent  back- 
ground. The  prints  should  be  made  on  a  rough  paper  so 
that  they  can  be  written  on ;  blue-print  paper  is  excellent. 
This  method  is  applicable  to  hard  parts  which  may  be  studied 
dry;  e.g.,  mollusc  shells,  echinoderms,  various  large  arthnx 
pods,  epidermal  markings  of  vertebrates  and  parts  of  the 
vertebrate  skeleton.  Shadow  photographs  may  be  made  of 
the  outlines  of  opaque  objects,  such  as  birds'  bills,  birds'  eggs, 
and  butterfly  wings,  by  using  parallel  rays  of  light  and  inter- 
posing the  object  between  the  source  of  light*  and  the  photo- 

*  A  Welsbach  burner  or  an  electric  light  are  especially  good.    Minute 


MEASUREMENT   OF    ORGANISMS.  3 

graphic  paper.  More  or  less  transparent  organs,  such  as 
leaves,  petals,  insect-wings,  and  appendages  of  the  smaller 
Crustacea,  may  be  reproduced  either  directly  on  blue-print 
paper  or  by  "  solar  prints,"  either  of  natural  size  or  greatly 
enlarged.  For  solar  printing  the  objects  should  be  mounted 
in  series  on  glass  plates.  They  may  be  fixed  on  the  plate  by 
means  of  balsam  or  albumen  and  mounted  between  plates  either 
dry  or  in  Canada  balsam  or  other  permanent  mounting  media. 
Wings  of  flies,  orthoptera,  neuroptera,  etc.,  may  be  prepared 
for  study  in  this  way;  twenty-five  to  one  hundred  sets  of  wings 
being  photographed  on  one  sheet  of  paper,  say  16  X  20  inches 
in  size.  Microphotographs  will  sometimes  be  found  service- 
able in  studying  small  organisms  or  organs,  such  as  shells  of 
Protozoa  or  cytological  details. 

Camera  drawings  are  a  Convenient  although  slow  method  of 
reproducing  on  paper  greatly  enlarged  outlines  of  microscopic 
characters,  such  as  the  form  and  markings  of  worms  and 
lower  Crustacea,  sponge  spicules,  bristles,  scales  and  scutes, 
plant-hairs,  cells  and  other  microscopic  objects.  In  making 
such  camera  drawings  a  low-power  objective,  such  as  ZeissA*, 
will  often  be  found  very  useful. 

The  Determination  of  Integral  Variates.— 
Methods  of  Counting. 

While  the  counting  of  small  numbers  offers  no  special  diffi- 
culty, the  counting  becomes  more  difficult  with  an  increase  of 
numbers.  To  count  large  numbers  the  general  rule  is  to  di- 
vide the  field  occupied  by  the  numerous  organs  into  many 
small  fields  each  containing  only  a  few  organs.  Counting 
under  the  microscope,  e.g.,  the  number  of  spines,  scales  or 
plant-hairs  per  square  millimetre,  may  be  aided  by  cross-hair 
rectangles  in  the  eyepiece.  The  number  of  blood-corpuscles 
in  a  drop  of  blood,  or  of  organisms  in  a  cubic  centimetre  of 
water,  have  long  been  counted  on  glass  slides  ruled  in  small 
squares. 

electric  lamps  such  as  are  fed  by  a  single  cell  give  sharp  shadows  of 
small  objects. 


STATISTICAL   METHODS. 


The   Determination  of  Graduated  Variates.— 
Methods  of  Measurement. 

Straight  lines  on  a  plane  surface  are  easily  meas- 
ured by  means  of  a  measuring-scale  of  some  sort.  The  meas- 
urement should  always  be  metric  because 
this  is  the  universal  scientific  system.  Vari- 
ous kinds  of  scales  may  be  obtained  of 
optical  companies  and  hardware  dealers, — 
such  as  steel  measurin-g  tapes,  graduated  to 
millimetres  (about  $1.00),  and  steel  rules 
(6  cm.  to  15  cm.)  graduated  to  £  of  a  milli- 
metre. Steel  "spring-bow"  dividers  with 
milled-head  screw  are  useful  for  getting 
distances  which  may  be  laid  off  on  a  scale. 
Tortuous  lines,  e.g.,  the  contour  of  the 
serrated  margin  of  a  leaf  or  the  outer 
margin  of  the  wing  of  a  sphinx  moth,  may 
be  measured  by  a  map-measurer  ("Eutfer- 
nungsmesser,"  Fig.  1),  supplied  at  artist's 
and  engineer's  supply  stores  at  about  $8.50. 
Distances  through  solid  bodies 
or  cavities  are  measured  by  calipers  of 
some  sort.  Calipers  for  measuring  diameters 
of  solid  bodies  are  made  in  various  styles. 
Micrometer  screw  calipers  ("speeded") 
reading  to  one-huudredths  of  a  millimetre 
and  sold  by  dealers  in  physical  apparatus  for 
about  $5.00  are  excellent  for  determining  diameters  of  bones, 
birds'  eggs,  gastropod  shells,  etc.  Leg  calipers  for  rougher 
work  can  be  obtained  for  from  30  cents  to  $4.00.  The 
micrometer  "  caliper-square,"  available  for  inside  or  outside 
measurements  and  measuring  to  hundredths  of  a  millimetre, 
is  a  useful  instrument.* 

The  area  of  plane  surfaces,  as,  e.g.,  of  a  wing  or  leaf, 
is  easily  determined  by  means  of  a  sheet  of  colloidin  scratched 
in  millimetre  squares.  By  rubbing  in  a  little  carmine  the 

*  Many  of  the  instruments  described  in  this  section  are  made  by  the 
Starrett  Co.,  Athol,  Mass.,  and  by  Brown  and  Sharpe,  Providence,  tool 
cutters. 


FIG.  1. 


MEASUREMENT    OF   ORGAKISMS. 


scratches  may  be  made  clearer.  The  number  of  squares 
covered  by  the  surface  is  counted  (fractional  squares  being 
mentally  sum  mated)  and  the  required  area  is  at  once  obtained. 
If  the  area  has  been  traced  on  paper  it  may  be  measured  by 
the  plauimeter  (Fig.  2).  This  instrument  may  be  obtained  at 


FIG.  2. 

engineer's  supply  shops.  It  consists  of  two  steel  arms  hinged 
together  at  one  end;  the  other  end  of  one  arm  is  fixed  by  a 
pin  into  the  paper,  the  end  of  the  second  arm  is  provided  with 
a  tracer.  By  merely  tracing  the  periphery  of  the  figure  whose 
area  is  to  be  determined  the  area  may  be  read  off  from  a  drum 
which  moves  with  the  second  arm.  This  method  is  less 
wearisome  than  the  method  of  counting  squares. 

The  area  of  a  curved  surface,  like  that  of  the  elytra 
of  a  beetle  or  the  shell  of  a  clam,  is  not  always  easy  to  find. 
To  get  the  area  approximately,  project  the  curved  surface  on 
a  plane  by  making  a  camera  drawing  or  photograph  of  its 
outline.  By  means  of  parallel  lines  divide  the  outline  draw- 
ing into  strips  such  that  the  corresponding  parts  of  the  curved 
surface  are  only  slightly  curved  across  the  strips,  but  greatly 
curved  lengthwise  of  the  strips.  Measure  the  length  of  each 
plane  strip  and  divide  the  magnitude  by  the  magnification  of 
the  drawing.  Measure  also,  with  a  flexible  scale,  the  length 
of  the  corresponding  strip  on  the  curved  surface.  Then,  the 
area  of  any  strip  of  the  object  is  to  the  area  of  the  projection 
as  the  length  of  the  strip  on  the  object  is  to  the  length  of  its 
projection.  The  sum  of  the  areas  of  the  strips  will  give  the 
total  area  of  the  surface. 


6  STATISTICAL  METHODS. 

Characters  occupying  three  dimensions  of 
space  may  be  quantitatively  expressed  by  volume.  The 
volume  of  water  or  sand  displaced  may  be  used  to  measure 
volume  in  the  case  of  solids.  The  volume  of  water  or  sand  con- 
tained will  measure  a  cavity.  Irregular  form  is  best  measured 
by  getting,  either  by  means  of  photography  or  drawings,  pro- 
jections of  the  object  on  one  or  more  of  the  three  rectangular 
fundamental  planes  of  the  organ,  and  then  measuring  these 
plane  figures  as  already  described.  Or  two  or  more  axes  may 
be  measured  and  their  ratio  found. 

Characters  having  weight  are  easily  measured  ;  the 
only  precautions  being  those  observed  by  physicists  and 
chemists. 

Color  Characters.  Color  may  be  qualitatively  ex- 
pressed by  reference  to  named  standard  color  samples.  Such 
standard  color  samples  are  given  in  Ridgeway's  book, 
"  Nomenclature  of  Color,"  and  also  in  a  set  of  samples  manu- 
factured by  the  Milton  Bradley  Co.,  Springfield,  Mass. ,  costing 
6  cents.  The  best  way  of  designating  a  color  character  is  by 
means  of  the  color  wheel,  a  cheap  form  of  which  (costing  6 
cents)  is  made  by  the  Milton  Bradley  Co.  The  colors  of  this 
"top"  are  standard  and  are  of  known  wave-length  as  follows: 
Bed,  656  to  661  Green,  514  to  519 

Orange,  606  to  611  Blue,     467  to  472 

Yellow,  577  to  582  Violet,  419  to  424. 

It  is  desirable  to  use  Milton  Bradley's  color  top  as  a  standard. 
Any  color  character  can  be  matched  by  using  the  elementary 
colors  and  white  and  black  in  certain  proportions.  The  pro- 
portions are  given  in  percents.  In  practice  the  fewest  possible 
colors  necessary  to  give  the  color  character  should  be  employed 
and  two  or  three  independent  determinations  of  each  should 
be  made  at  different  times  and  the  results  averaged.  So  far 
as  my  experience  goes  any  color  character  is  given  by  only 
one  least  combination  of  elementary  colors.  (See  Science, 
July  16,  1897.) 

When  there  is  a  complex  color  pattern  the  color  of  the 
different  patches  must  be  determined  separately.  In  case  of 
a  close  intermingling  of  colors,  the  colored  area  may  be  rapidly 
rotated  on  a  turntable  so  that  the  colors  blend  and  the  result- 


MEASUREMENT   OF   ORGANISMS.  7 

ant  may  then  be  compared  with  the  color  wheel.  By  this 
means  also  the  total  melanism  or  albinism,  viridescence,  etc., 
may  be  measured. 

Marking-characters.  The  quantitative  expression  of 
markings  or  color  patterns  will  often  call  for  the  greatest 
ingenuity  of  the  naturalist.  Only  the  most  general  rules  can 
here  be  laid  down.  Study  the  markings  comparatively  in  a 
large  number  of  the  individuals,  reduce  the  pattern  to  its 
simplest  elements,  and  find  the  law  of  the  qualitative  variation 
of  these  elements.  The  variation  of  the  elements  can  usually 
be  treated  under  one  of  the  preceding  categories.  Find  in  how 
far  the  variation  of  the  color  pattern  is  due  to  the  variation  of 
some  number  or  other  magnitude,  and  express  the  variation  in 
terms  of  that  magnitude.  Remember  that  it  is  rarely  a  ques- 
tion whether  the  variation  of  the  character  can  be  expressed 
quantitatively  but  rather  what  is  the  best  method  of  express- 
ing it  quantitatively. 

Aids  in  Calculating.  An  indispensable  aid  in  multi- 
plying and  dividing  is  a  book  of  reckoning  tables  of  which 
Crelle's  Rechnungstafeln  (Berlin:  Geo.  Reimer)  is  the  best. 
This  work  enables  us  to  get  directly  any  product  to  999  X  999 
and  indirectly,  but  with  great  rapidity,  any  higher  product  or 
any  quotient. 

The  tables  of  Barlow  ("  Tables  of  Squares,  Cubes,  Square 
Roots,  Cube  Roots,  and  Reciprocals  of  all  Integer  Numbers 
up  to  10,000")  are  like  our  Table  X,  but  more  extended. 

The  tedious  work  of  adding  columns  of  numbers  is  greatly 
simplified  by  the  use  of  some  one  of  the  better  adding  ma- 
chines. There  are  many  forms;  of  which  the  best  are  made 
in  the  United  States.  The  author  has  used  the  "  Comp- 
tometer" made  by  the  Felt  and  Tarrent  Manufacturing  Co., 
Chicago  ($225),  and  found  it  perfectly  satisfactory.  This 
machine  is  manipulated  by  touching  keys,  as  in  a  typewriter, 
but  it  does  not  print  the  numbers  touched  off.  In  this  respect 
it  is  inferior  to  the  Burroughs  Adding  Machine  of  the  Ameri- 
can Arithometer  Co.,  St.  Louis,  Mo.,  which  costs  $250  to  $350, 
or  to  the  Standard  Adding  Machine,  St.  Louis  ($185). 

For  the  multiplication  and  division  of  large  numbers  the 
Baldwin  Calculator  is  well  spoken  of  (Science,  xvn,  706).  It 
is  sold  by  the  Spectator  Company,  95  William  Street,  New 
York,  price  $250.  The  same  firm  is  agent  for  Tate's  Im- 


8  STATISTICAL    METHODS. 

proved  Arithometer  ($300  to  $400).  The  "Brunsviga"  cal- 
culating machine  (Herrn  Grimme,  Natalia  &  Co.,  Brunswick, 
Germany,  Manufacturers;  price  $- 140  to  v 75)  is  highly  recom- 
mended by  Pearson. 

To  draw  logarithmic  curves  and  for  the  mechanical  solu- 
tion of  arithmetical  problems  the  instrument  of  Brooks 
(Science,  xvn,  690,  not  yet  marketed)  should  be  found  useful. 

Precautions  in  Arithmetical  Work.  Even  the 
most  careful  computers  make  mistakes  in  arithmetical  wrork. 
It  is  absolutely  necessary  to  take  such  precautions  that  errors 
may  be  detected.  The  best  method  is  for  statistical  workers 
to  compute  in  pairs,  but  absolutely  independently,  comparing 
results  as  the  work  progresses,  so  that  time  shall  not  be 
wasted  by  elaborate  work  done  with  erroneous  values.  In 
case  of  disagreement  botli  workers  should  recompute,  start- 
ing from  that  point  of  the  work  where  their  results  check.  In 
cases  where  it  is  not  feasible  for  the  work  to  be  done  by  two 
people,  it  should  be  calculated  on  distinct  pages  of  the  note- 
book— proceeding  through  several  steps  on  the  one  page  and 
then  independently  through  the  same  steps  on  another  page; 
checking  the  work  as  it  progresses.  It  will  be  found  useful 
as  the  work  progresses  to  make  rough  checks  by  comparing 
the  results  with  the  original  data  to  see  that  the  results  are 
probable. 

Neatness  in  arrangement  of  work  and  in  the  making  of 
figures  is  essential.  It  is  best  to  make  all  calculations  in  a 
book  with  pages  about  20  cm.  by  30  cm.,  quadruple  ruled, 
with  about  three  squares  to  the  centimetre,  so  that  each 
figure  may  occupy  a  distinct  square.  I  like  to  work  with  a 
pencil,  of  2H  grade,  so  that  slight  errors  may  be  erased  and 
rectified.  In  case  of  larger  errors  running  through  several 
steps  of  the  work,  the  erroneous  calculations  should  not  be 
erased  but  cancelled. 

In  using  logarithms  with  the  six-place  table  given  in  this 
book,  it  is  ordinarily  necessary  to  write  the  entire  mantissa 
to  six  places,  and  to  determine  the  number  corresponding  to 
any  logarithm  to  at  least  six  places  by  use  of  the  table  of 
proportional  parts  given  at  the  bottom  of  the  page.  Upon 
the  completion  of  the  calculation  the  number  of  decimal 
places  to  be  recorded  will  depend  upon  the  probable*  error  of 


MEASUREMENT   OF   ORGANISMS.  0 

each  constant.  It  will  ordinarily  suffice  if  the  probable  error 
contain  two  significant  figures,  e.g.,  ±0.17  or  ±O.OOS9;  then 
the  constant  will  be  carried  out  to  the  same  number  of  places 
and  not  farther. 


iQ  STATISTICAL   METHODS. 


CHAPTER  II. 

ON  THE  SERIATION  AND  PLOTTING  OF  DATA  AND  THE 
FREQUENCY  POLYGON. 

The  data  obtained  by  measuring  any  character  in  a  lot  of 
individuals  consists  either  of  amass  of  numbers  for  the  charac- 
ter in  each  individual ;  or,  perhaps,  two  numbers  which  are  to 
be  united  to  form  a  ratio  ;  or,  finally,  a  series  of  numbers  such 
as  are  obtained  by  the  color  wheel,  of  the  order  :  W  40$,  N 
(Black)  38$,  7  12$,  O  101  The  first  operation  is  the  simplifi- 
cation of  data.  Each  variate  must  be  represented  by  one 
number  only.  Consequently,  quotients  of  ratios  must  be  de- 
termined and  that  single  color  of  a  series  of  colors  which  shows 
most  variability  in  the  species  must  be  selected,  e.g.,N. 

The  process  of  seriation,  which  comes  next,  consists  of  the 
grouping  of  similar  magnitudes  into  the  same  magnitude 
class.  The  classes  being  arranged  in  order  of  magnitude, 
the  number  of  variates  occurring  in  each  class  is  determined. 
The  number  of  variates  in  the  class  determines  the  frequency 
of  the  class.  Each  class  has  a  central  value,  an  inner  and  an 
outer  limiting  value,  and  a  certain  range  of  values. 

The  method  of  seriation  may  be  illustrated  by  two  examples  ;  one  of 
integral  variates,  and  the  other  of  graduated  variates. 

Example  1.  The  magnitude  of  21  integral  variates  are  found  to  be  as 
follows  :    12,  14,  11,  13,  12,  12,  14,  13,  12,  11,  12,  12,  11,  12,  10,  11,  12, 13,  13, 
13,  12,  12.    In  seriation  they  are  arranged  as  follows  : 
Classes:         10,11,12,13,14. 
Frequency :    1,    4, 11,    4,    2. 

Example  2.  In  the  more  frequent  case  of  graduated  variates  our  mag- 
nitudes might  be  more  as  follows  : 

3.2  4.5  5.2  5.6  6.0      . 
3.8           4.7           5.2           5.7  6.2 
4.1           4.9           5.3           5.8  6.4 

4.3  5.0  5.3  5.8  6.7 
4.3           5.1           5.4           5.9           7.3 

In  this  case  it  is  clear  that  our  magnitudes  are  not  exact,  but  are  merely 
approximations  of  the  real  (forever  unknowable)  value.  The  question 


SERIATION  AND   PLOTTING   OF   DATA. 


11 


arises  concerning  the  inclusiveiiess  of  a  class— the  class  range.  An 
approximate  rule  is  :  Make  the  classes  only  just  large  enough  to  have 
no  or  very  few  vacant  classes  in  the  series.  Following  this  rule  we  get 

3.0-3.4;    3.5-3.9;    4.0-4.4;    4.5-4.9;    5.0-5.4; 
Classes.... 

Frequency 


Classes.... 


Frequency 

The  classes  are  named  from  their  middle  value,  or  better,  for  ease  of 
subsequent  calculations,  by  a  series  of  small  integers  (1  to  9). 

In  case  the  data  show  a  tendency  of  the  observer  towards  estimating 
to  the  nearest  round  number,  like  5  or  10,  each  class  should  include  one 
and  only  one  of  these  round  numbers. 

As  Fechner  ('97)  has  pointed  out,  the  frequency  of  the  classes  and  all 
the  data  to  be  calculated  from  the  series  will  vary  according  to  the 
point  at  which  we  begin  our  seriation.  Thus  if,  instead  of  beginning  the 
series  with  3.0  as  in  our  example,  we  begin  with  3.1  we  get  the  series  : 

[  3.1-3.5;    3.6-4.0;    4.1-4.5;    4.6-5.0;    5.1-5.5; 


3.2 

3.7 

4.2 

4.7 

5.2 

1 

2 

3 

4 

5 

1 

1 

3 

3 

7 

5.5-5.9; 

6.0-6.4; 

6.5-6.9; 

Y.0-7.4; 

5.7 

6.2 

6.7 

7.2 

6 

7 

8 

9 

5 

3 

1 

1 

Classes.... 
Frequency 
Classes .... 


3.3 

1 


(  5.6-6.0; 
(      5.8 


3.8 
1 

6.1-6.5; 
6.3 


4.3 

4 

6.6-7.0; 
6.8 
1 


4.8 


3.5 
6 


7.1-7.5; 

7.3 

Frequency  6211 

which  is  quite  a  different  series.  Fechner  suggests  the  rule:  Choose  such 
a  position  of  the  classes  as  will  give  a  most  normal  distribution  of  fre- 
quencies. According  to  this  rule  the  first  distribution  proposed  above 
is  to  be  preferred  to  the  second. 

In  order  to  give  a  more  vivid  picture  of  the  frequency  of 
the  classes  it  is  important  to  plot  the  frequency  polygon. 
This  is  done  on  coordinate  paper.* 

The  best  method,  especially  when  the  number  of  classes 
is  less  than  20,  is  to  represent  the  frequencies  by  rectangles 
of  equal  base  and  of  altitude  proportional  to  the  frequencies. 
Lay  off  along  a  horizontal  line  equal  contiguous  spaces  each 
of  which  shall  represent  one  class,  number  the  spaces  in  order 
from  left  to  right  with  the  class  magnitudes  in  succession, 
and  erect  upon  these  bases  rectangles  proportionate  in  height 
to  the  frequency  of  the  respective  classes  (Fig.  3). 


*  This  paper  may  be  obtained  at  any  artists'  supply  store. 


12  STATISTICAL   METHODS. 

This  method  of  drawing  the  frequency  polygon  is  known  as 
the  method  of  rectangles. 

When  the  number  of  classes  is  large  the  frequencies  may  be 
represented  by  ordinates  as  follows :   At  equal  intervals  along 


~n    ,    , 

,     ,     1 

3.0         3.5         4.0        4.5         5.0         5.5         6.0         6.5         7.0         7.5 

FIG.  3. 

a  horizontal  line  (axis  of  X)  draw  a  series  of  (vertical)  ordi- 
nates whose  successive  heights  shall  be  proportional  to  the 
frequency  of  the  classes.  Join  the  tops  of  the  ordinates  as 
shown  in  Fig.  4.  This  method  of  drawing  the  frequency 
polygon  is  known  as  the  method  of  loaded  ordinates. 


2600  LEAVES 


-NORMAL  CURVE 


600 
6CO 
620 
480 

440 

i 

>•? 

s 

2 

\, 

'/ 

\, 

/ 

^ 

7 

i 

\ 

, 

Q400 

I 

j 

s 

Z 

\ 

/ 

i 

^ 

2 

1; 

i 

\ 

^"00 

\x> 

H 
' 

V 

120 

- 

\\ 

/ 

\ 

40 

/ 

i 

\ 

^- 

'    - 

^5. 

9         10         11         12         13         1-i 
NUMBER  OF  VEINS 

15 

10      u 

MEAN 

18 

19 

20 

21        22"' 

23  J 

FIG.  4. — VEINS  IN  BEECH  LEAVES,  AFTER  PEARSON,  '02'. 

The  rejection  of  extreme  variates  in  calculating 
the  constants  of  a  distribution  polygon  is  to  be  done  only 
rarely  and  with  caution.  In  many  physical  measurements 
Chauvenet's  criterion  is  used  to  test  the  suspicion  that  a 
single  extreme  variant  should  be  rejected.  A  limiting  devia- 
tion (KG)  is  calculated.  K  is  the  argument  in  Table  IV  cor- 
responding to  a  tabular  entry  equal  to  -  —  • 

471 


SERIATIOK   AND   PLOTTING   OF   DATA.  13 

EXAMPLE. — In  1000  minnows  from  one  lake  there  are  found  the 
following  frequencies  of  anal  fin-rays: 

789  10  11  12  13 

1  2  15  279  554  144  5 

A  =  10.835 ;     a  =  .728  fin-rays. 
1999       .__„ 

K=:*ooo==-49975- 

Looking  in  Table  IV  we  find  3.48  corresponding  to  the  entry  49975. 
Then  the  limiting  deviation  =  3. 48  X. 728  =  2.5334  and  the  limiting  class 
is  10.835  —  2.533=8.302;  hence  the  observation  at  7  might  be  excluded 
in  calculating  the  constants  of  the  seriation ;  but  it  should  not  be  sup- 
pressed in  publishing  the  data. 

CERTAIN  CONSTANTS  OF  THE  FREQUENCY  POLYGON. 

After  the  data  have  been  gathered  and  arranged  it  is  neces- 
sary to  determine  the  law  of  distribution  of  the  variates.  To 
get  at  this  law  we  must  first  determine  certain  constants. 

The  average  or  mean  (A)  is  the  abscissa  of  the  centre  of 
gravity  of  the  frequency  polygon.  It  is  found  by  the  formula 


in  which  V  is  the  magnitude  of  any  class;  /  its  frequency; 
2  indicates  that  the  sum  of  the  products  for  all  classes  into 
frequency  is  to  be  got,  and  n  is  the  number  of  variates. 

Thus  in  the  example  on  p.  10: 
A  =(3.2X1+3.7X1+4.2X3+4.7X3  +  5.2X7+5.7X5+6.2X3 

+  6.7  X  1  +7.2 XI)-*- 25  =  5.24, 
or 
AI=*  (IX 1+2  XI  +3X3  +  4X3  +5X7  +6X5  +7X3+8X1+  9X1) 

-*- 25=5.08, 
A  =  5.2*  +  .08(5.7-5.2)  =  5.24. 

A  still  shorter  method  of  finding  A  is  given  on  page  20. 

The  mode  (M)  is  the  class  with  the  greatest  frequency. 
It  is  necessary  to  distinguish  sharply  between  the  empirical 
and  the  theoretical  mode.  The  empirical  mode  is  that  mode 
which  is  found  on  inspection  of  the  seriated  data.  In  the 
example,  the  empirical  mode  is  5.2.  The  theoretical  mode  is 
the  mode  of  the  theoretical  curve  most  closely  agreeing  with 
the  observed  distribution.  Pearson  1902b,  p.  261)  gives  this 

*  5.2  is  the  true  class  magnitude  corresponding  to  the  integer  5. 


14  STATISTICAL   METHODS. 

rule  for  roughly  determining  the  theoretical  mode.  The 
mode  lies  on  the  opposite  side  of  the  median  from  the  mean  ; 
and  the  abscissal  distance  from  the  median  to  the  mode  is 
double  the  distance  from  the  median  to  the  mean;  or, 
mode=mean  —  3 X  (mean  —  median).  More  precise  directions 
for  finding  the  mode  in  the  different  types  of  frequency  poly- 
gons are  given  in  the  discussion  of  the  types. 

The  median  magnitude  is  one  above  which  and  below 
which  50%  of  the  variates  occur.  It  is  such  a  point  on  the 
axis  of  X  of  the  frequency  polygon  that  an  ordinate  drawn 
from  it  bisects  the  polygon  of  rectangles  or  the  continuous 
curve,  but  not  the  polygon  of  loaded  ordinates. 

To  find  its  position:  Divide  the  variates  into  three  lots:  those  less  than 
the  middle  class,  i.e.,  the  one  that  contains  the  median  magnitude,  of 
which  the  total  number  is  a;  those  of  the  middle  class,  b;  and  those 
greater,  c.  Then  a  +  b  +  c  =  n  =  the  total  number  of  variates:  Let  I'  = 
the  lower  limiting  value  of  the  middle  class,  and  I"  =the  upper  limiting 
value,  and  let  x  =  the  abscissal  distance  of  the  median  ordinate  above  the 
lower  limit  or  below  the  upper  limit  of  the  median  class  according  as  x 
is  positive  or  negative.  Then  \n  —  a  :  b  =  x  :  I"  —  V  when  x  is  positive, 
or  %n  —  c  :  b  =  x  :  I"  —  V  when  x  is  negative. 

Thus  in  the  last  example:  (12.5-8):  7  =x  :  0.5;  #=.32;.  the  median 
magnitude  =  5.0 +  .32  =  5.32.  Or  (12.5-10):  7  =  -x  :  0.5;  z=-.18; 
the  median  magnitude  =5. 5 -.18  =  5. 32.  (Cf.  p.  10.) 

The  probable  error  (E)  of  the  determination  of 

any  value  gives  the  measure  of  unreliability  of  the  determina- 
tion; and  it  should  always  be  found.  For,  any  determination 
of  a  constant  of  a  frequency  polygon  is  only  an  approximation 
to  the  truth.  The  probable  error  (E)  is  a  pair  of  values  lying 
one  above  and  the  other  below  the  value  determined.  We 
can  say  that  there  is  an  even  chance  that  the  true  value  lies 
between  these  limits.  The  chances  that  the  true  value  lies 
within  :* 

±2£are  4,5:1  ±5E  are  1,310:1 

±3Eare21  :1  ±6E  are  19,200:1 

±4E  are  142  :1  ±7E  are  420,000:1 

±8£are  17,000,000:1 

±9E  are  about  a  billion  to  1. 

The   probable   error   should   be  found   to   two   significant 
*  These  values  are  easily  deduced  from  Table  IV. 


SEKIATIOST   AND   PLOTTING   OF   DATA.  15 

figures.  The  determination  of  which  it  is  the  error  should 
be  carried  out  to  the  same  number  of  places  as  the  probable 
error  and  no  more. 

The  probable  difference  between  two  averages  (Al  and 
•A  2)  of  which  the  probable  errors  (El  and  E2)  are  known  is 
the  square  root  of  the  sum  of  the  squared  probable  errors,  or 
(Pearson,  '02): 

Probable  Difference  of  Al-A2  is  \/E*  +  E*. 

The  probable  error  of  the  mean  is  given  by  the 
formula 

tion  [gee  bglow] 


Vnumber  of  variates  \/n 

It  will  be  seen  that  the  probable  error  is  less,  that  is,  that 
the  result  is  more  accurate,  the  greater  the  number  of  variates 
measured,  but  the  accuracy  does  not  increase  in  the  same  ratio 
as  the  number  of  individuals  measured,  but  as  the  square  root 
of  the  number.  The  probable  error  of  the  mean  decreases  as 
the  standard  deviation  decreases. 

The_  probable  error  of  the  median  is  ±.84535<r 
+\/n  (Sheppard,  '98). 

The  geometric  mean  of  a  series  of  values  (v)  is  the 
number  corresponding  to  the  average  of  the  logarithms  of 
the  values.  Thus, 

J_(log^) 

\JT  —  IV  • 

n 

The  index  of  the  variability,  a,  of  the  variates  when 
they  group  themselves  about  one  mode  is  found  by  adding 
the  products  of  the  squared  deviation-from-the-mean  of  each 
class  multiplied  by  its  frequency,  dividing  by  the  total 
number  of  variates,  and  extracting  the  square  root  of  the 
quotient,  thus: 


V 


sum  of  [(deviation  of  class  from  mean)2 

X  frequency  of  class] 

number  of  variates 


where  X  is  the  number  of  units  in  the  class  range,  frequently 
unity. 


16  STATISTICAL   METHODS. 

This  measure  is  known  as  the  standard  deviation.  It 
is  a  concrete  number  expressed  in  the  units  of  the  classes. 
This,  the  best  measure  of  variability,  is  expressed  geomet- 
rically as  the  half  parameter,  or  the  abscissa  of  the  point  on 
the  frequency  curve  where  the  change  of  curvature  (from 
concave  to  convex  toward  the  centre)  occurs. 

The  probable  error  of  the  standard  deviation  is 

+  n.B74K      ^andard  deviation      _  ±0-6745    * 
V2  X  number  of  variates  \/2n 

Other  Indices  of  Variation.  The  average  deviation, 
or  average  departure,  is  found  thus: 

_       sum  of  [deviations  of  class  from  mean  X  frequency] 
number  of  variates 

The   average   deviation  is  equal  to    .7  97  9  X  standard   deviation,   or 


The  probable  (or  mid)  departure  is  the  distance  from  the  mode 
of  that  ordinate  which  exactly  bisects  the  half  curve  QMX  or  OMX1, 
Fig.  5,  it  is  equal  to  0.6745  X  standard  deviation  =  0.6745o.  Neither 
of  these  last  two  indices  of  variation  is  as  good  as  the  standard  devia- 
tion when  n  is  rather  small. 

The  standard  deviation,  like  the  other  indices  of  variation, 
is  a  concrete  number,  being  expressed  in  the  same  units  as 
the  magnitudes  of  the  classes.  The  standard  deviation  of 
one  lot  of  variates  is  consequently  not  comparable  with  the 
S.  D.  of  variates  measured  in  other  units.  It  has  been  pro- 
posed to  reduce  the  index  of  variation  to  an  abstract  number, 
independent  of  any  particular  unit,  by  dividing  the  index  of 
variation  of  any  variates  by  the  mean;  the  quotient  multi- 
plied by  100  is  called  the  coefficient  of  variability.  In 

a  formula,  C=-^XlOO%  (Pearson,  '96;    Brewster,  '97). 

The  probable  error  of  the  coefficient  of  vari- 
ability is  given  by  Pearson  as: 


SERIATIOX   AND    PLOTTING    OF    DATA.  17 


When  C  is  small,  say  less  than  10%,  the  factor  in  brackets 
may  be  omitted,  especially  as  only  two  significant  figures 
of  the  probable  error  need  be  recorded. 

The  average,  standard  deviation,  coefficient  of  correlation, 
and  their  probable  errors  may  be  conveniently  calculated  al- 
together by  logarithms,  as  shown  in  the  paradigm  on  page  38. 

QUICK  METHODS  OF  ROUGHLY  DETERMINING  AVERAGE  AND 
VARIABILITY.* 

1.  Arrange  the  specimens  in  a  series  according  to  the  mag- 
nitude of  the  character,  simply  judging  the  order  by  the  eye. 
Then  pick  out  those  two  that  will  divide  the  series  into  thirds 
and  measure  them.  Their  average  will  be  the  average  of  the 
whole  series.  Then, 

Mean  — the  smaller  of  the  two  measures  _ 
.43 

(.43  is  the  value  of  ±  — ,  at  which  the  area  of  the   curve 

included  between  these  limits  of  x  equals  one-third  of  the 
whole) . 

Or,  2.  Select  roughly  two  specimens  that  seem  to  be  about 
one-third  of  the  distance  from  the  two  extremes  £nd  group 
all  others  as  larger  than  the  larger  one,  smaller  than  the 
smaller  one,  or  between  the.  two.  Measure  the  two  speci- 
mens. Count  the  number  in  each  group  and  determine  o 
by  aid  of  Table  IV  (p.  120)  as  follows:  Taking  as  origin  the 
middle  of  the  whole  series,  call  the  number  of  leaves  from 
the  middle  to  the  smaller  n2.  and  the  number  from  the 
middle  to  the  larger  w2".  Also,  the  x  distance  to  the  lower 
division  point  h^  and  to  the  upper  division  point  h2.  Then 
(/i1+/i2)  =  the  range  covered  by  the  middle  division  or  the 
difference  between  the  upper  and  lower  value.  As  we  know 
the  areas  of  the  curve  between  the  origin  and  h^  on  the  one 
hand  and  h2  on  the  other  (percentage  of  individuals  between 

the  middle  and  h,  and  h2),  we  can  find  —  and  —  from  Table  IV, 

a  a 

x 
since  they  are  the  values  —  corresponding  to  the  percentage 

*  See  Macdonell,  1902. 


18  STATISTICAL   METHODS. 


areas  determined.     But  —  +— =•  :     thus  a  is  deter- 

G      a  a 

mined.  Knowing  a  we  can  get  h^  or  h2,  and  hence  the  mean. 
Or  the  value  of  the  character  of  the  middle  specimen  may 
be  taken  as  the  mean  value. 

EXAMPLE. — Seventy-six  beech-leaves  which  had  fallen  from  one 
tree  were  picked  up.  They  were  sorted  out  as  in  the  second  method. 
It  was  found  that  22  were  smaller  than  the  smaller  type  leaf,  which 
was  1.78  inches  in  length;  and  23  were  larger  than  the  larger  type  leaf 
(2.22  inches  in  length).  The  38th  leaf  is  the  middle  of  the  series,  and 
so  the  smaller  type  leaf  was  distant  16  leaves  from  the  middle,  and 
the  larger  15. 


From  Table  IV: 


.21223 

.20884 


Therefore  ^  =  .555. 


Similarly      =  .5 


h2 


A!  =  .2278,  A2=-2122. 

Mean  is  at  1.78  +  .2278  =2.01. 


THE   CLASSES    OF   FREQUENCY    POLYGONS,         19 


CHAPTER   III. 
THE  CLASSES  OF  FREQUENCY  POLYGONS. 

The  plotted  curve  may  fall  into  one  of  the  following'classes : 

A.  Unimodal. 

I.  Simple. 

1.  Range  unlimited  in  both  directions: 

a.  Symmetrical.     The  normal  curve. 

b.  Unsym metrical  (Pearson's  Type  IV). 

2.  Range  limited   in  one    direction,    together    with 

skewness  (Types  III,  V,  and  VI). 

3.  Range  limited  in  both  directions  : 

a.  Symmetrical,  Type  II. 

b.  Unsymmetrical,  Type  I. 
II.  Complex. 

B.  Multimodal. 

The  classification  of  any  given  curve  is  not  always  an  easy 
task.  Whether  the  curve  is  unimodal  or  multimodal  can  be 
told  by  inspection.  Whether  any  unimodal  curve  is  simple 
or  complex  cannot  be  told  by  any  existing  methods  without 
great  labor  and  uncertainty  in  the  result. 

Complex  curves  may  be  classified  as  follows  : 

1.  Composed  of  two  curves,  whose  modes  are  different  but  so  near  that 
the  component  curves  blend  into  one  ;   such  curves  are  usually  unsym- 
metrical. 

2.  The  sum  of  two  curves  having  the  same  mode  but  differing  varia- 
bility. 

3.  The  difference  of  two  curves  having  the  same  mode  but  differing 
variability. 

If  the  material  is  believed  to  be  homogeneous  and  the  curve 
is  unimodal  it  is  probably  simple  and  its  classification  may  be 
carried  further. 

For  classification  the  rule  is  as  follows  :  Determine  the  mean 
of  the  magnitudes.  Take  a  class  near  the  mean  (call  it  Fo) 


20  STATISTICAL  METHODS. 

as  a  zero  point  ;  then  the  departure  of  all  the  other  classes 
will  be  -  1,  -  2,  -  3,  etc.,  and  +  1,  +  2,  -f  3,  etc. 

Add  the  products  of  all  these  departures  multiplied  by  the 
frequency  of  the  corresponding  class  and  divide  by  n\  call 
the  quotient  VL 

Add  the  products  of  the  squares  of  all  the  departures  multi- 
plied by  the  frequency  of  the  corresponding  class  and  divide 
by  n\  call  the  quotient  v*. 

Add  the  products  of  the  cubes  of  all  the  departures  multiplied 
by  the  frequency  of  the  corresponding  class  and  divide  by  n\ 
call  the  quotient  vz. 

Add  the  products  of  ike  fourth  powers  of  all  the  departures 
multiplied  by  the  frequency  of  the  corresponding  class  and 
divide  by  n\  call  the  quotient  v±.  Or, 

—  y  } 

—  —  =  departure    of    FQ  from   mean.      F0   being 


known,  A  may  be  found  [A 


The  values  rlt  r3)  r$,  r4,  are  called  respectively  the  first, 
second,  third,  and  fourth  moments  of  the  curve  about  Fo. 

To  get  the  moments  of  the  curve  about  the  mean,  either  of 
two  methods  (A  or  B)  will  be  employed.  Method  A  is  used 
when  integral  variates  are  under  consideration  ;  method  B 
when  we  deal  with  graduated  variates. 

(A)  To  find  moments  in  case  of  integral  variates: 


*  This  is  the  short  method  of  finding  A  referred  to  on  page  13. 


THE  CLASSES  OF  FREQUENCY  POLYGONS.    21 


71 

^5  ~  5"i»4  +  lOVva  -  lOVv.,  +  4y15  ; 


(B)  To  find  moments  in  case  of  graduated  variates: 


y^  -  10V*2  +  4y15  -  f  /£JA5; 
in  which  A  is  the  class  range  expressed  in  the  same  unit  as 
the  average. 


The  probable  error  of  the  preceding  constants  in  the  special 
case  of  the  normal  curve  is  as  follows: 

Efi2=  .67449<724/-  ;  E^=  .67449 

71  ' 

=  .6T4494     -; 

T       71 

=  •67449V//i<r  (p-  31)  ; 

#  of  Skewness=  .67449y  7^.     (See  page  30.) 
(From  Pearson,  1903C). 

The  classification  of  any  empirical  frequency  polygon 
depends  upon  the  value  of  its  "  critical  function,"  F*  (Pear- 
son, 1901d). 


4(4A-3/?1)(2/?2-3A-6)< 


*  This  value  of  F  is  general.     For  the  special  case  of  Types  I-IV 
the  following  critical  function  was  given  by  Pearson  and  has  been 


STATISTICAL   METHODS. 


Value  of  F. 


Corresponding  Frequency  Curve. 


F>1  and  <oo 
F=l 

F>0and  <1 
F-O/A-OtA-S 

F=Q,  ft  =  0,  02  not  =  3 
.F<0 


Type  III.     Transitional  between  Type 

I  and  Type  VI. 
Type  VI. 
Type  V.     Transitional  between  Type 

IV  and  Type  II. 
Type  IV. 
Normal  curve. 
Type  II. 
Type  I. 


An  important  relation  to  be  referred  to  later  is- 


3A-2&  +  6' 
M 

f 

\ 

1 

\ 

I 

\ 

1 

\ 

1 

\ 

1 

\ 

1 

\ 

/ 

\ 

• 

/ 

\ 

/ 

i  i 

5. 

\ 

&      4      3       2      1       v 

FIG. 

\.      2      . 

5      1     2 

THE  NORMAL  CURVE. 

The  normal  curve  is  symmetrical  about  the  mode;  con- 
sequently the  mode  and  the  median  and  mean  coincide. 
The  mathematical  formula  of  the  normal  curve,  a  formula 

much  used.     Fi  =  2p2—  3/?i  —  6.     The  classification  was  given  as  follows: 

,,  .  ,.  ..    |  /?i>0,  curve  is  of  Type  I. 

When  F  u  negative  and  ^  ^  ^  ^  ^  .g  rf  Typ<j  „ 

-|  *>»•  *>»•  CUrVe  Is  °f  Typf  IIL 

<  /?i=0,  ^2  =  3,  curve  is  normal. 

When  F  is  positive  and  ft>  0,  52>  3,  curve  is  of  Type  IV, 


When  . 


and 


THE  CLASSES  OF  FREQUENCY  POLYGONS.    23 

of  which  one  does  not  have  to  understand  the  development 
in  order  to  make  use  of  it,  is 


This  formula  gives  the  value  of  any  ordinate  y  (or  any  class) 
at  any  distance  x  (measured  along  the  base,  X,  X',  of  Fig.  5) 
from  the  mode,  e  is  a  constant  number,  2.71828,  the  base 
of  the  Naperian  system  of  logarithms,  n  is  the  total  area 
of  the  curve  or  number  of  variates,  and  o  is  the  Standard 
Deviation,  which  is  constant  for  any  curve  and  measures  the 
variability  of  the  curve,  or  the  steepness  of  its  slope. 

To  compare  any  observed  curve  with  the  theo- 
retical normal  curve  we  can  make  use  of  tables.  For 
the  case  of  a  polygon  of  loaded  ordinates  the  theoretical  fre- 

quency of  any  class  at  a  deviation  —  from  the  mean  can  be 

taken  directly  from  Table  III.     Here  —  is  the  actual  devia- 

o 

tion  from  the  mean  expressed  in  units  of  the  standard  devia- 

tion, and  3L  the   corresponding  ordinate,   yQ  being  taken  as 

2/o 
equal  to  1,  and  <r  is  the  standard  deviation. 

For  the  case  of  a  polygon  built  up  of  rectangles  represent- 
ing the  relative  frequency  of  the  variates,  Table  IV  gives 
immediately  the  theoretical  number  of  individuals  occurring 

between  the  values  z=0  and  x=  ±—  .     By  looking  up  the 

x 
given  values  of  —  the  corresponding  theoretical  percentage 

G 

of  variates  between  the  limits  rc=0  and  x=  ±—  will  be  found 

a 

directly.     The  ratio  —  may  be  called  the  Index  of  Abmodality. 

The  normal  curve  may  preferably  be  employed  even  when 
/?!  is  not  exactly  equal  to  0,  nor  /?2  exactly  equal  to  3,  nor  F 
exactly  equal  to  0.  Use  the  normal  curve  when 


and 


24  STATISTICAL   METHODS. 

also  the  skewness  (p.  30)  should  be  less  than  twice  the  value 
.67449    /- 


To  determine  the  closeness  of  fit  of  a  theoreti- 
cal polygon  to  the  observed  polygon.  Find  for 
each  class  the  difference  (<^)  between  the  theoretical  value  (y) 
and  the  observed  frequency  (/).  Divide  the  square  of  this 
difference  in  each  case  by  y.  The  square  root  of  the  sum  of  the 

/~d~2 
quotients  is  the  index  of  closeness  of  fit  (/).     Or,  A=  4/  I—  • 

a 

The  probability  (P:l)  that  the  observed  distribution  is  truly 
represented  by  the  theoretical  polygon  may  be  calculated  from 
the  following  formula,  to  use  which  the  number  of  classes 
(A)  must  be  odd  or  must  be  made  odd  by  the  addition  of  a 
class  with  0  frequency. 


__ 
2      2.42.4.6 


4£ZL_\ 
.4.6...,l-3/' 


This  is  the  method  of  Pearson,  1900&. 

To  determine  the  probability  of  a  given  dis- 
tribution being  normal.  Having  found,  in  units  of  the 
standard  deviation,  the  deviation  (7)  of  the  inner  limiting 
value  (L)  of  each  class  from  the  average,  look  up  the 
corresponding  class-index  a  from  Table  IV.  Or,  better,  find  a 
directly  for  each  class  by  dividing  the  half  of  the  total  num- 
ber of  variates  minus  all  those  lying  beyond  the  inner  limit- 
ing value  of  the  class  in  question  by  the  half  of  the  total 

I  */ 

number  of  variates;  or,  in  a  formula,  -y—  -;  where  JT0Y/  means 

?n 

add  all  the  frequencies  from  the  median  value  to  ^,  and  n 
is  the  number  of  variates.  Next  find  for  each  class  the  sum 
of  A-}-  a%.  This  should  equal  L.  The  difference  is  the 
actual  discrepancy.  The  probable  discrepancy  should  next  be 
calculated  for  all  but  the  extreme  values.  It  is  calculated 
by  use  of  the  formula 


0.6745, 


THE  CLASSES  OF  FREQUENCY  POLYGONS. 


where  the  value  of  z  corresponding  to  %  is  got  from  Table  III, 
or  from  the  formula 

1 


The  ratio  of  actual  to  probable  discrepancy  is  next  to  be 
calculated  for  each  class.  The  probable  limit  (P.L.)  of  the 
ratios  varies  with  the  number  (A)  of  ratios  found,  according 
to  the  following  table : 


Ai 

P.L. 

A\ 

P.L. 

Ai 

P.L 

Ai 

P.L. 

I 
2 
3 
4 
5 

1.000 
1.559 
1.874 
2.088 
2.248 

6 
7 
8 
9 
10 

2.375 

2.481 
2.570 
2.648 
2.716 

11 
12 
13 
14 
15 

2.777 
2.832 
2.882 
2  928 
2  970 

16 
17 
18 
19 
20 

3.009 
3.046 
3.080 
3.112 
3.142 

The  foregoing  method  is  from  Sheppard  (1898). 

The  probable  range  of  abscissae  (2xt)  of  a  normal  dis- 
tribution, or  that  beyond  which  the  theoretical  frequency  (y) 
is  less  than  1,  varies  with  the  number  of  variates  (ri)  as  well 
as  with  a,  in  accordance  with  the  following  formula  derived 

by  the  transposition  of  y=  — -p=-e~x2/2<r*  by  putting  y=l: 

a\/2x 


V  l~6 


2xt=  2(7 


Example.  For  the  ventricosity  of  1000  shells  of  Lit- 
tornea  littorea  from  Tenby,  Wales,  A  =  90.964%  and  o= 
2.3775%.  What  is  the  probable  range  of  ventricosity 
expressed  in  per  cent.? 


/ 
. 


1  000 


The  observed  range  was  15  (Duncker,  '98).  See  also  the 
criterion  of  Chauvenet  ('88)  for  the  rejection  of  extreme 
variates  (page  12). 

THE  NORMAL  CURVE  OF  FREQUENCY  AS  A  BINOMIAL 
CURVE. 

The  normal  curve  may  also  be  expressed  by  the  binomial 
formula  (pXq)A,  where  p=J,  g=i,  and  A  is  the  number  of 


26  STATISTICAL   METHODS. 

terms,  less  1,  in  the  expansion  of  the  binomial;  hence  approx- 
imately the  number  of  classes  into  which  the  magnitudes  of 
the  variates  should  fall.  If  the  standard  deviation  be  known, 
A  may  be  found  by  the  equation 

J=4X  (Standard  Deviation) 2=4<r2. 


Example  of  Normal  Curve. — Number  of  rays  in  lower  valve 
of  Pecten  opercularis  from  Firth  of  Forth: 


V 

14 
15 
16 
17 
18 
19 
20 
21 


/ 

F-FO 

f(V—Vo) 

/(F-T 

1 

-3 

-3 

9 

8 

-2 

-16 

32 

63 

-1 

-63 

63 

154 

0 

0 

0 

164 

1 

164 

164 

96 

2 

192 

384 

20 

3 

60 

180 

2 

4 

8 

32 

=  508 


342 


864 


-27 

-64 

63 

0 

164 
768 
540 
128 

1446 


81 

128 

63 

0 

164 

1S36 

1620 

512 

4104 


A  =  F0+  vi  =  17  +.6732  =  17.6732.     „ 

/m2=  1.7008 -0.67322  =1.2475;  <r  =  V^  =1.1169. 

^  =2.8465  -3  X0.6732  Xl.7008  +2  X0.67323  =0.0217. 

/x4  =8.0787  -4  X0.6732  X2.8465  +6X0.67322  Xl.7008  -3  X0.67324 

0  02172  4  4223  =4.4223. 

U.u^J.*  r\  (\f\f\n.     a     -z.-m&'J QA~\  A 

2  ~  i   1  i  £»7To  =^.o414;. 


-0.00047 ;     F/jL23  =  0.0009. 


0.0002  X5.84142 


4XH.3650X(-0.3178) 
3v>2-2v14_3(1.7008)2-2X.70594_ 
v4  "L8.0787 


508 


Theoretical  maximum  frequency,  2/0  = — 

cV2«     1.1169V2^ 


=  181.5. 


The  probable  discrepancy,  based  on  the  five  larger  values 
of  y,  is  found  as  follows,  the  Xi  values  being  taken  from  a 
tablelike  Table  IV: 


L 

a 

Xi 

A  +<r\i 

Actual 
Dis- 
crepancy. 

Probable 
Dis- 
crepancy. 

Ratio  of 
Actual  to 
Probable 
Dis- 

14.5 

-0.99606 

crepancy. 

15.5 

-0.96457 

-2.11 

15.34 

+  0.17 

.083 

2.05 

16.5 

-0.71654 

-1.07 

16.51 

-0.01 

.032 

0.31 

17.5 

-0.11023 

—  0.138 

17.55 

-0.05 

.025 

2.00 

18.5 

+  0.53543 

0.73 

18.51 

—0.01 

.027 

0.37 

19.5 

+  0.91439 

1.72 

19.62 

-0.12 

.054 

2.22 

20.5 

+  0.99213 

THE  CLASSES  OF  FREQUENCY  POLYGONS.    2? 

The  extreme  values  are  not  calculated  for  the  relations 
indicated  by  the  formula  do  not  hold  well  there  where  the 
frequencies  are  small  and  the  proportionate  values  of  y  are 
changing  rapidly  for  small  changes  of  x  For  the  five  values 
considered  the  actual  discrepancy  is  less  than  the  probable 
discrepancy  in  three  cases  and  less  than  the  probable  limit 
in  all. 

To  find  the  average  difference  between  the 
pt\\  and  the  (pi  l)th  individual  in  any  seriation 
(Gait oil's  difference  problem).  Let  xp  be  the  aver- 
age interval  between  the  pth  and  (p-fl)th  individual;  n  the 
total  number  of  variates;  and  a  their  standard  deviation. 

Then,  (1)  when  n  is  large  and  p  small: 


.  _  V2np    pVe'V 


P_  nym* 

where  ym—     / — e~^m  . 

m  can  be  found  from  Table  IV  by  the  use  of  the  formula 


where  the  value  of  m  sought   is   the   argument  correspond- 
ing to  the  tabular  entry    (  --  —  j  . 


-=?-  +  1.875^2.  (^V; 

ym  n3          \ym] 


.- 
-p)p  n1      y 

57_± 

ym 


28  STATISTICAL   METHODS. 

_ocfa-p)5  +  p5     ~r(n-pY-p*     w_ 
-^    -*nn-          ' 


31m*-101raM-28 
2/m 

The  solution  of  the  equations  for  cuc2,  and  c3  will  be  facili- 
tated by  finding,  once  for  all,  the  logarithms  of  n,  (n  —  p), 

tn 
(n-2p))(n-p)p,  and—  . 

i/m 

(2).  When  n  and  />  are  both  large  and  not  nearly  equal: 


(3).  When  n  is  small  the  unsimplified  form  of  the  equa"- 
tion  must  be  used. 


X--  (1  +  ^  +  ^+03+  ...)• 

ym 

\n  means  the  products  of  all  integers  from  1  to  n.  The 
series  clt  c2,  c3  is  not  complete,  but  the  values  of  c  with  higher 
subscripts  are  so  small  that  they  may  be  neglected. 

Let  Ip'p"  be  the  difference  measured  in  units  of  o  between 
the  p'th  and  the  p"th  individual,  then 


The  foregoing  method  is  that  of  Pearson  (1902k)  based 
upon  some  considerations  of  Galton  (1902). 

To  find  the  best  fitting  normal  frequency  dis- 
tribution when  only  a  portion  of  an  empirical 
distribution  is  given. 

First  apply  the  following  parabola  of  the  second  order: 


THE  CLASSES  OF  FREQUENCY  POLYGONS.    29 

2  1 


+£ij+s*\T)  \> 

where  Z  is  the  half  range  and 


2/0          07    1         A)         **'-2         i  r£2?         A  7  J*         *1**;Z    72' 

61  1O  7/1^  77?  Ot 

To  find  m0  arrange  the  frequencies  in  the  usual  manner 
(p.  26)  and  find  the  logarithm  of  each;  their  sum  is  equal 
to  ra0.  Making  the  class  situated  at  the  middle  of  the 
range  0,  find  the  deviation  of  each  of  the  other  classes  from 
this  class.  The  algebraic  sum  of  the  product  of  the  loga- 
rithms by  the  deviations  gives  mr  The  second  moment 
about  the  same  zero  point  gives  m2.  Or, 


. 

Substituting  in  (1)  we  get  a  numerical  quadratic  equation 
which  can  be  put  in  the  form 


If  the  normal  curve  be  y=z0e 
(3)  y 


whence,  by  comparison  of  right-hand  expressions  in  equa- 
tions (2)  and  (3), 


(£  2\ 
'o-jjjjj 


Then  the  required  normal  curve  is 

i-vr*** 

(Pearson,  1902m.) 


30  STATISTICAL   METHODS. 

OTHER  UNIMODAL  FREQUENCY  POLYGONS. 
The  formulas  of  Pearson's  Types  I  to  VI  are  as  follows: 

Type     I.     y=4+Q 

/         X2\ 

Type   II.     y=y0(l—fi)    . 

Typelll.    y=y0(l  +  jYe~x/d' 

Type  IV.     y=y0cos62me~1:0,  where  tan  6=-r. 

Type    V.     y=y0x~pe~r/x. 

Type  VI.     y=y0(x-l)«*/3«*. 

In  these  formulas: 
rr,  abscissae; 
y0,  the  ordinate  at  the  origin,  to  be  especially  reckoned  for 

each  type; 
y,  the  height  of  the  ordinate  (or  rectangle)  located  at  the 

distance  x  from  2/0; 
I,  a  part  of  the  abscissa-axis  XX'  expressed  in  units  of  the 

classes; 
e,  the  base  of  the  Naperian  system  of  logarithms,  2.71828. 

The  other  letters  stand  for  relations  that  are  explained  in 
the  sections  below  treating  of  each  type  separately. 

The  range  of  the  curve  is  limited  in  both  directions  in 
Types  I  and  II,  is  limited  in  one  direction  only  in  Types  III, 
V,  and  VI,  and  is  unlimited  in  both  directions  in  Type  IV 
and  the  normal  curve.  The  normal  curve  may  give  the  best 
fit,  however,  notwithstanding  the  fact  that  in  biological 
statistics  the  range  is  ordinarily  limited  at  both  extremes. 
Thus  the  range  of  carapace  length  to  total  length  of  the 
lobster  is  limited  between  0  and  1.  The  ratio  of  carapace 
length  to  abdominal  length  in  various  crustaceans  may,  how- 
ever, conceivably  take  any  value  from  +  oo  to  0.  In  the  ratio 
of  dorso ventral  to  antero-posterior  diameter  the  forms  of  the 
molluscan  genera  Pinna  or  Mafleus  on  the  one  hand  and 
Solen  on  the  other  approach  such  extremes. 

Asymmetry  or  Skewiiess  (a)  is  found  in  Types  I,  III, 
IV,  V,  and  VI.  In  skew  curves  the  mode  and  the  mean  are 


THE  CLASSES  OF  FKEQUEKCY  POLYGONS.    31 

ated  from  each  other  by  a  certain  distance  D;   or  D  = 

an  —  mode.      Asymmetry  is  measured  by  the  ratio  a=  —  . 

a 

If  the  mean  is  greater  than  the  mode,  skewness  is  positive; 
if  the  mean  is  less  than  the  mode,  skewness  is  negative.  D, 
and  hence  skewness,  may  be  calculated  when  the  theoretical 
mode  is  known  (see  pages  13,  14,  and  below). 

In  Types  I  and  III  skewness   is    measured  also   by  the 

raf.o       -jVSjff,      where        '-%£&       When 

5/92  —  6/?t  —  9  is  positive,  a  has  the  sign  of  /*3;  if  negative, 
a  has  the  opposite  sign  to  /(3  (Duncker,  '00b). 

InType      i. 

"      "       III,  a=%\/Ji=  —  ±fi/3—,  where  the  sign  is  the 
+  2V  1*2*  same  as  that  of  /*3. 


P 
since  p  —  4  is  the  positive  root  of  the  quadratic: 


p  is  readily  found. 
InType    VI,  a= 

(ql 

where  (1—qJ  and  (q2+l)  are  the  two  roots  of  the  equation 


To  compare  any  observed  frequency  polygon 
of  Type  I  with  its  corresponding  theoretical 
curve. 

2/= 


32  STATISTICAL   METHODS. 

To  find  119  12,  ml9  m2,  yQ.  * 

The  total  range,  Z,  of  the  curve  (along  the  abscissa  axis] 
is  found  by  the  equation 


1= 

Zj  and  12  are  the  ranges  to  the  one  side  and  the  other  of  t/0; 

2.  a; 


™1  .  m™2 


=s  —  2; 


To  solve  this  equation  it  will  be  necessary  to  determine 
the  value  of  each  parenthetical  quantity  following  the  F 
sign  and  find  the  corresponding  value  of  F  from  Table  V. 
It  is,  however,  sometimes  easier  to  calculate  the  value  of  yQ 
from  the  following  approximate  formula: 


_n    (m1  +  m2+l)\m1  +  m2  12 
I  \/27tmlm2 


_J1___L\ 
2    m\    m2/ 


With  these  data  the  theoretical  curve  of  Type  I  maybe 
drawn.  Frequency  polygons  of  Type  I  are  often  found  in 
biological  measurements. 

To  compare  any  observed  frequency  polygon 
of  Type  II  with  its  corresponding  theoretical 
curve. 


This  equation  is  only  a  special  form  of  the  equation  of  Type 
I  in  which  ZX=Z2  and  m1  =  m2. 

As  from  page  22,  ^  =  0  in  Type  II,  l=2a\/s+l;  since  the 
curve  is  symmetrical,  D—  0,  and 

t/      ON  n    F(m+1.5) 

m=i(«-2);    7/0=-—  A—      -f-. 


The  F  values  will  be  found  from  Table  V. 


THE   CLASSES    OF   FREQUENCY    POLYGONS.         33 

An  approximate  formula  for  yQ  is  given  by  Duncker  as  fol- 
lows: 

s-1 


.  _  - 

\/(*  +!)(*-  2) 


4(8-2). 


To  compare  any  observed  frequency  polygon 
of  Type  III  with  its  corresponding  theoretical 
curve. 

X\  P   -x/d 

"//  e 

The  range  at  one  side  of  the  mode  is  infinite;  at  the  other 
is  found  by  the  formula 

l,=  a  ^^=  <2—^-  (for  Type  III). 
2\/ft 

I,        I,  n          "p+l 

Also,  p=j-=  —  ;     yQ=- 


j\          j 
D      aa' 


7--  —        —  • 
I,     epr(p+l) 


The  value  of  F  corresponding  to  p+l  can  be  got  from 
Table  V,  Appendix. 

To  compare  any  observed  frequency  polygon 
of  Type  IV  with  its  corresponding  theoretical 
curve. 

This  is  the  commonest  type  of  biological  skew  curves. 


6  is  &  variable,  dependent  upon  x  as  shown  in  the  equation 


The  factor  (cos  6)2m  following  y0  indicates  that  the  curve 
is  not  calculated  from  the  mean  ordinate  (A),  or  the  mode 
(A  —  D),  but  that  the  zero  ordinate  is  at  A  —mD;  or  at  a  dis- 
tance mXD  from  the  mean. 


the  opposite  sign  to 


34  STATISTICAL   METHODS. 


0  (arc  of  circle)  =       5,  • 


nA/s   e 
2/o=y4/ oT- 


___ 


<j>= angle  whose  tangent  is  — . 
s 

To  compare  any  observed  frequency  polygon 
of  Type  V  with  its  corresponding  theoretical 
curve. 

To  find  p  solve  the  quadratic  equation 
and  take  the  positive  root. 

r=a(p -2)v/Pz:3;    ?/o=rrLiv    D==S^^Y 


To  compare  any  observed  frequency  polygon 
of  Type  VI  with  its  corresponding  theoretical 
curve. 

y=y0(x-W2/*qi- 

1  —  ql  and  q2-{- 1  are  the  two  roots  of  the  equation 


li=s  i^-'  <,  where  (1-qJ  and  s  are  negative; 

- 


D=; 


*  The  foregoing  value  is  approximate  and  is  applicable  when,  as  is 
usually  the  case,  a  is  greater  than  2.  The  exact  value  is  given  by 
Pearson  as 


/„  > 

(sin  6feT0dd 
. 


the  formula  for  reducing  which  is  to  be  gained  from  the  integral  cal- 
culus. 


THE    CLASSES    OF    FREQUENCY    POLY<i()\S.          35 

Example  of  calculating  tlie  theoretical  curve  corre- 
sponding vvitli  observed  data.  (Fig.  6.) 

Distribution  of  frequency  of  glands  in  the  right  fore  leg  of  ~COO  female 
swine  (integral  variates): 

Number  of  glands      0123456789       10 
Frequency 15      209    365    482    414    277    134     72      22      8         2 

Assume  the  axis  yy'  ( Vni)  to  pass  through  ordinate  4,  then: 
V        V-Vm       f     /(F-Fm)    f(V—  F«>)2    /(F- F»*)3    /(F  —  Fm)« 


0 

4 

15 

—  60 

240 

—  960 

3840 

1 

3 

209 

—  627 

1881 

—  5643 

16929 

2 

o 

365 

—  730 

1460 

—  2920 

5840 

3 

—  1 

482 

—  482 

482 

—  482 

482 

4 

0 

414 

0 

0 

0 

0 

5 

1 

277 

277 

277 

277 

277 

6 

2 

134 

268 

536 

1072 

2144 

7 

3 

72 

216 

648 

1944 

5832 

8 

4 

22 

88 

352 

1408 

5632 

9 

5 

8 

40 

200 

1000 

5000 

10 

6 

o 

12 

72 

432 

2592 

2        2000  —998  6148  —3872  48568 

Vl  =  —  •   998  -*-  2000  =  —    .499. 

ra  =       6148  -*-  2000  =       3.074. 

v,  =  —  3872  -*•  2000  =  —  1.936. 

v4  =     48568  -*-  2000  =     24.284. 

/n=0;  A  =  4-.  499  =3.501. 

Ma  =  3.074  —  (—  .499)2  =  2.824999. 

M3  =  -  1.936  -  3(-  .499  X  3.074)  +  2(-  .499)'  =  2.417278. 

M4  =  24.284  -4(-.  499  X-  1.936)  -f  6(.249001  X  3.074)  -  3(-  499)*  =  24.826297, 

(2.417278)*  _    5.843232929  _ 
Pl       (2.824999)3  "  22.545241683  ~ 
_   24.826297  _  24.826297  _ 

^^ 


_ 
(2.824999^  -  7.98061935 

.259  X  (6.111)2 

4(12.443  -  .778)(6.222  -  6.778) 
6(3.11082-  0^25918  - 
.555o9 


,21.9857 


a-  K  V.259178  ~^-  =  .31115. 

D-  1.680774  X  .3111  =  .5230. 
Z).a-  .5230  X  19.9857  =  10.4519. 

Z=  .840387  4/16  X  20.9857  -f  0.25918  X  (21.9857)2  =  18.0448. 
18.0448-  10.4519  =  ^^ 


36  STATISTICAL   METHODS. 

Ja= 18.0448  -3.7965  =  14.2483; 
3.7965X17.9857 


-         18.0448 
14.2483X17.9857 
1^0448  - 


. 
378401' 


17.9846      vo1_1QO_.0833(.0556-.2643-J0704) 

A  4.L 


18.0448  \/2  K  x  3.7840  X  14.2006 

=  475.24,  the  frequency  of  the  modal  class. 

Position  of  the  mode,  '  y0=*A  -D=3.50l -.523  =  2.978.  The  close- 
ness of  fit  to  the  theoretical  curve  is  calculated  below  by  Pearson's 
method  (page  24). 


V  f  Theoretical  (y}  d  d* 

-1  0  0.0  0.0 

0  15  21.1  -   6.1  37.21  1.76 

1  209  185.8  +23.2  538.24  2.90 

2  365  395.1  -30.1  906.01  2.30 

3  482  475.2  +   6.8  46.24  .10 

4  414  405.6  +  8.4  70.56  .17 

5  277  272.1  +   4.9  24.01  .09 

6  134  147.6  -13.6  184.96  1.25 

7  72  65.9  +   6.1  37.21  .57 

8  22  24.1  -   2.1  4.41  .18 

9  8  7.0  +   1.0  1.00  .14 

10  2  1.6  +  0.4  .16  .10 

11  0  0.2  -  0.2  .04 

12  0.0 

^—=9.56 

y 


That  is,  the  probability  is  that  in  one  out  of  every  two  random  series 
belonging  to  Type  I  we  should  expect  a  fit  not  essentially  closer 
than  that  given  by  our  series,  which,  of  course,  assures  us  that  this 
distribution  is  properly  classified  under  Type  I. 


THE  USE  OF  LOGARITHMS  IN  CURVE-FITTING. 

Most  of  the  statistical  operations  can  be  greatly  facilitated 
by  the  use  of  logarithms.     In  curve-fitting  their  use  becomes 


.01234.56T 

FIG.  6. 
Distribution  of  frequency  in  glands  of  swine. 

»     polygon  of  observed  frequency. 

—  -  —  -,  polygon  of  theoretical  frequency  (Type  I). 
-  •  -  -,  normal  frequency  polygon. 


38  STATISTICAL   METHODS. 

necessary.     The  following  paradigm  will  be  found  of  assist- 
ance: 

GENERAL. 

log  vj = log  2(V-  F0)  -  log  n.  A  =  Vm  +  vr 

log  v2=  log  2(V—  V0) 2  — log  n.  log  a=  J  log  fiy. 

log  i/3=  log  2(V-  F0)3  -log  n.  log  C=  J  log  /*2-log  A. 

Iogv4=log  ^'(F-F0)4  -logn. 

log  E.  A  =  9.828982  +  log  a  -  J  log  n. 

log  E.a  =  log  E.A  -  0. 150515. 

logE.c=logE.ff-log  A. 

log  2=  0.301030  TV=  .08333        Find  2  log  ^ 

log  3=  0.477121  ^=.02916  3  log  vl 

log  4=0.602060  Tjf  0  =  .0125  4  log  vx 

log  6=0.778151          log  i= 9.  98970 

V2=N(\og  v2)  -AT (2  log  Vl)  -[.0833].      Find:  log  /£2;  2  log  /z2; 

3  log  M2. 
j«3=  AT(log  v3)  -  JV(log  3  +  log  v,  +  log  v2)  +  AT  (log  2  +  3  log  vx) 

Find:   log  /£,;   2  log  py 
fi4=  N(\og  y4)  -TV (log  4  +  log  yt  +  log  v3) 

+  AT  (log  6  +  2  log  vj  +  log  v2)  -  A^dog  3  +  4  log  vj 
-7V[9.698970  +  log  ^2]-^.     Find  log  /i4. 
log  /?!=  2  log  /£3  -  3  log  /£2. 
log/?2=log//4-21og/M2. 
T(;==5/32-6/3i-9  (Types  I,  IV). 

Skewness: 

Type      I:  log  «=1  .og  ft  +  log  w- log  (ft  +  3)  -0.301030. 
Type  III:  log  a=J  log  ft-0.301030. 


THE    CLASSES   OF    FREQUENCY    POLYGONS.         39 

Type   IV:  log  a=  J  log  ft  +  log  (/?2-f  3)  -  log  w  -  0.301030. 

Type     V:  log  a=log  2  +  i  log  (p-3)-log  p. 

Type   VI:  log  a=log  (&  +  &)  +  J  log  (ft-&-3)-log  (&-&) 


TYPE  IV. 


This  is  the  most  difficult  of  all  the  types  to  be  fitted.  The 
work  of  fitting  is  carried  out  by  the  use  of  logarithms,  as 
follows  : 


log  /=i  log  ft  +  log  («-2).     log  fc=l 

log  a=log  /-log  (s  +  2)  -0.301030. 

log  1=  J  log  /£2  +  i  log  {  N[log  (*-!)  +  1  .204120] 

-JV[log  A  +  2  log(s-2)]|  -0.602060. 

s  +  2 


log  ml)  =  log  A:  -0.602060. 

log  r=log  /b  +  log  s-  0.602060  -log 

log  tan  6—  log  T  —  log  s. 


—  log  3s) 

—N  (8.920819  -log  s)-N(log  r+log  ^)]  +  9.637784) 
-0.399090-log  Z-(s  +  l)  log  cos  0. 
log  y=logy0+N  [log  (s  +  2)  +  log  log  cos  6] 

T]. 


MULTIMODAL   CURVES. 

Multimodal  curves  are  given  when  the  frequency  in  the 
different  classes  exhibits  more  than  one  mode.  False  mul- 
timodal  curves  result  from  too  few  observations,  or  when  the 
classes  are  too  numerous  for  the  variates.  By  increasing  the 
number  of  variates  or  by  making  the  classes  more  inclusive 
some  of  the  modes  disappear. 


*  Tn  degrees  and  fractions  of  a  degree;    see  Table  VII. 


40  STATISTICAL   METHODS. 

Multimodal  curves  differ  in  degree.  The  modes  may  be  so 
close  that  only  a  single  mode  (usually  in  an  asymmetrical 
curve)  appears  in  the  result;  or  one  of  the  modes  may  appear 
as  a  hump  on  the  other;  or  the  two  modes  may  even  be  far 
apart  and  separated  by  a  deep  sinus  (Figs.  7  to  10). 


6.5   4.5   3.5   2.5    1.5    .5  0.5     1.5  2.5    3.5   4.5   5.5  6.5    7.5 
FIG.  7. 

Pearson  has  offered  a  means  of  breaking  up  a  compound 
curve  with  apparently  only  one  mode  into  two  curves  having 
distinct  modes;  but  this  method  is  very  tedious  and  rarely 
applicable. 


2 
FIG.  8. 

The  index  of  divergence  of  two  modes  of  a  multi- 
modal  curve  is  the  distance  between  the  modes  expressed  in 


THE    CLASSICS    OF    FREQUENCY    POLYGONS.         41 


terms  of  the  standard  deviation  of  the  Linore  variable  of  the 
components.* 

The  index   of  isolation   of    two  masses  of  variates 
grouped  about  adjacent  modes  is  the  ratio  of  the  depression 
between  the  modes  to  the  height  of  the  shorter  mode. 
,.    The  meaning  of  multimodal  curves  is  diverse.     Sometimes 


FIG.  9. 

they  indicate  a  polymorphic  condition  of  the  species,  the  modes 
representing  the  different  type  forms.     This  is  the  case  with 


3      i 


321012 

FIG.  10. 

the  number  of  ray  flowers  of  the  white  daisy  which  has  modes 
at  8,  13,  21,  34,  etc.  Sometimes  they  indicate  a  splitting  of  a 
species  into  two  or  more  varieties. 

*  I  have  proposed  (Science,  VII,  685)  to  measure  the  divergence  in  a 
unit  =3 X Standard  Deviation,  which  has  certain  advantages  >n  species 
itudy. 


42  STATISTICAL   METHODS. 


CHAPTER  IV. 
CORRELATED  VARIABILITY. 

Correlated  variation  is  such  a  relation  between  the  magni- 
tudes of  two  or  more  characters  that  any  abmodality  of  the 
one  is  accompanied  by  a  corresponding  abmodality  of  the 
other  or  others. 

The  methods  of  measuring  correlation  given  below  are 
applicable  to  cases  where  the  distribution  of  variates  is 
either  symmetrical  or  skew. 

The  principles  upon  which  the  measure  of  correlated  varia- 
tion rests  are  these.  When  we  take  individuals  at  random  we 
find  that  the  mean  magnitude  of  any  character  is  equal  to  the 
mean  magnitude  of  this  character  in  the  whole  population. 
Deviation  from  the  mean  of  the  whole  population  in  any  lot  of 
individuals  implies  a  selection.  If  we  select  individuals  on 
the  basis  of  one  character  (A ,  called  the  subject)  we  select  also 
any  closely  correlated  character  (B,  called  the  relative)  (e.g., 
leg-length  and  stature).  If  perfectly  correlated,  the  index  of 
abmodality  (p.  23)  of  any  class  of  B  will  be  as  great  as  that  of 
the  corresponding  class  of  A ,  or 

Index  abmodality  of  relative  class  _ 
Index  abmodality  of  subject  class  ~~ 

If  there  is  no  correlation,  then  whatever  the  value  of  the 
index  of  abmodality  of  the  subject,  that  of  the  relative  will 
be  zero  and  the  coefficient  of  correlation  will  be 

Index  of  abmodality  of  relative  class  _   0  _ 
Index  of  abmodality  of  subject  class  ~~  m  ~    ' 

The  coefficient  of  correlation  is  represented  in  formulas  by 
the  letter  r.  We  cannot  find  the  degree  of  correlation  be- 
tween two  organs  by  measuring  a  single  pair  only;  it  is  the 
correlation  "in  the  long  run"  which  we  must  consider. 
Hence  we  must  deal  with  masses  and  with  averages. 


CORRELATED    VARIABILITY. 


CI»-iOOOOi-<C»<?* 

I  I  I  I 


lO   SO 
CO    CO 


•  8  8  8  '•£•  3 


s  z  s 


*-*  o  o  o  o 

I  I  I 


as  s 


I-    CO    Tt»    O    ~ 

OS    GO    GO    i— I    ^« 


O   »-i   CM   CO   CO 


I 


Cl   C*   »-i 


CO    »i    «O    I-    CO    CO 


oo  QO  »o  co 


28  a 


C*   QO   rj«   QO 
O   »O   QO 


>  d 


I     I     I     I 


14  STATISTICAL  METHODS. 

In  studying  correlation  one  (either  one)  of  the  characters  is 
regarded  as  subject  and  the  other  as  relative.  A  correlation 
table  is  then  arranged  as  in  the  example  on  page  43,  which 
gives  data  for  determining  the  correlation  between  the  num- 
ber of  Mulleriaii  glands  on  the  right  (subject)  and  left  (rela- 
tive) legs  of  male  swine.  The  selected  subject  class  is  called 
the  type;  the  corresponding  distribution  of  the  relative  mag- 
nitudes is  called  the  array. 

METHODS  OF  DETERMINING  COEFFICIENT  OF  CORRELATION. 

Gallon's  graphic  method.  On  co-ordinate  paper 
draw  perpendicular  axes  JTand  T  \  locate  a  series  of  points 
from  the  pairs  of  indices  of  abmodality  of  the  relative  and  sub- 
ject corresponding  to  each  subject  class.  The  indices  of  the 
subjects  are  laid  off  as  abscissae  ;  the  indices  of  the  relatives 
as  ordinates,  regarding  signs.  Get  another  set  of  points  by  mak- 
ing a  second  correlation  table,  regarding  character  B  as  subject 
and  character  A  as  relative.  Then  draw  a  straight  line  through 
these  points  so  as  to  divide  the  region  occupied  by  them  into 
halves.  The  tangent  of  the  angle  made  by  the  last  line  with 
the  horizontal  axis  XX  (any  distance  yp,  divided  by  xp)  is  the 
index  of  correlation. 

A  more  precise  method  is  given  by  Pearson  as  follows: 
Sum  of  products  (deviation  subj.  class  X  deviation  each  assoc. 

r_el.  class  X  PP.  of  cases  in  both) 

total  no.  of  indivs.  X  Stand.  Dev.  of  subject  x  Stand.  Dev. 
of  relative ; 

or,  expressed  in  a  formula  : 

2  (dev.  x  X  dev.  y  X  /) 

T  = 

n<Ti<Ji 

This  method  requires  finding  many  products  in  the  numera- 
lor,  as  many  sets  of  products  as  there  are  entries  in  the  body  of 
the  correlation  table.  A  portion  of  the  products  to  be  found 
in  correlation  table,  p.  43,  is  indicated  below: 

(-  3.540  X  8 

-  3.547  X   <-  2.540  X  5 

(-  1.540  X  2 

f-  3.540  X      4 
"1.540  X  161 
,540  X    58 
etc* 


-  2.547  X  «(  ~  2.540  X  161 

|-i.r- 


CORRELATED    VARIABILITY.  45 


The  handling  of  long  decimal  fractions  may  be  avoided  by 
the  use  of  a  method  similar  to  that  used  at  page  26  for  find- 
ing the  average  and  standard  deviation.  The  formula  for  r 
may  be  written 


Assuming  the  class  including  or  nearest  to  the  true  mean 
of  the  subject  values  as  the  mean  of  the  subjects,  and  the 
class  including  or  nearest  to  the  true  mean  of  the  relative 
values  as  the  mean  of  the  relatives,  find  for  each  variate  the 
product  of  its  deviations  xf  and  yf  from  the  respective  assumed 
means,  and  (having  regard  for  signs)  find  the  algebraic  sum 
of  these  products.  Divide  this  sum  by  the  number  of  vari- 
ates;  the  quotient  is  the  average  of  the  deviation  products  about 
the  assumed  axes.  To  refer  to  the  true  axes,  passing  through 
the  true  means,  find  the  average  moments,  v1  (as  on  page  26), 
both  for  the  subject  and  the  relative  distributions  about  their 
respective  assumed  means,  and  subtract  the  product  of  the 
two  values  of  vl  from  the  average  of  the  approximate  devia- 
tion products  already  found.  Divide  the  difference  by  the 
product  of  the  standard  deviations  of  the  two  frequency  dis- 
tributions. (Compare  Yule,  '97b,  pp.  12-17.) 

The  probable  error  of  the  determination  of  r  is 

E  =0.6745(l-r2) 

\/n 


(Pearson  and  Filon,  '98,  p.  242.) 

Example.  Correlation  in  number  of  Miillerian  glands 
on  right  and  left  legs  of  2000  male  swine.  (See  table  on  next 
page.) 

For  +  quadrants  Z(x'y')=   5243 
"    -          "          J  Cry)  =-118 

5125 

— 0=  2.5625= 


STATISTICAL    METHODS. 


5s           rfi       (N       r-t       CO 
N            (N       O       •*       •* 

b.       O       <M       cO       O       CO 
Oi       <N       O       »O       O       CO 
C<l       CO       |>       (N       CO 

(N    O 

CO    O 
CO    (M 

"^          N,         o     >o     co     i> 

CO               H             ®       £       g       § 
b» 

00            ^       CO       (N 

"H    00   t>.    O    ^   »O  CC 
r>.   1>   O    O    CO   (M  U2 

ca  GO  05  O  «o  cop 

^      |     C-)    •<*   i-KN 

C5iC 

i-l                      0 

1                                                                                                                                                     1                  1                   *     M        ' 

1                                                                                                   II          I     CO 

C^JT-i 

,                  TJ4         CO         <N         rH 

if       J         I      1      1      1 

rH         (N         CO         Tf         1C         CO                           *T          II     v     <N 
S'S' 

INI 

-     <0 

5c  bio 

•c  ^ 


COl>  ^_, 


rH    ^-O    ool>    ^05    £<N     SN 


w    co^-t    oO    ,-!>    SCO    ^^0 


CO    (NN    TOO    000 

" 


096 
^  cog     — 

IN  g«N  g^-i       c  g     OS 
w  t    OST 
o  8     6SI 


6     818 


^         -HCO 


CO  O5      rnrf      MOO      WH 

-l^  rH       |  (N      |  j 


,  r 


TOOT 

E2        1881  — 


o  ^6 

g  i-oet-    g 


'  Z-ZIQ-    ^ 


CORRELATED   VARIABILITY.  47 


-W    -i-  =(2.5625-.  4535  X.  4605) 


*L7T95X  1.730  a=s°-7911- 


A/2000 

The  average  variability  of  an  array  is  =<r\/l—  r2. 
The  coefficient  of  regression  marks  the  proportional 
change  of  the  relative  organ  for  a  unit's  change  of  the  sub- 

ject organ.     It  is  given  by  the  equation  p=r  —  ,  where  a^  is 

<72 

the  standard  deviation  of  the  subject,  o2  that  of  the  relative. 


THE  QUANTITATIVE  TREATMENT  OF  CHARACTERS  NOT  QUAN- 
TITATIVELY MEASURABLE. 

Even  qualities  that  do  not  lend  themselves  to  a  quanti- 
tative expression  may  be  expressed  in  a  roughly  quantitative 
fashion.  The  fundamental  assumption  is  made  that  the 
frequencies  would  obey  the  normal  law  of  frequency  more 
or  less  closely,  provided  a  quantitative  scale  could  be  found. 
This  assumption  will  not,  in  most  biological  data,  lead  us  far 
astray. 

Divide  the  data  into  three  classes  (e.g.,  in  eye-color  we  may 
have  black,  brown  and  gray,  and  blue),  and  let  the  frequency 
of  these  classes  be  n},  n2,  n.3,  in  which  nt  and  n.3  are  each  less 
than  \n,  so  that  n2  contains  the  median.  Let  Llf  L.f  be  the 
(unknown)  distances  of  the  mean  from  the  two  boundaries 
of  n2.  Call  Li/a=hl  and  L3/a=h3,  then 


7T./0 

and 

nr+n2-n3 
n 


/2  f 
V   n  JQ 


48 


STATISTICAL  METHODS. 


Now  the  left-hand  side  in  these  equations  is  known  ;    it  is  \a 
of  Table  IV.     From  this  table  the  right-hand  value  of  the 


FIG.  11. 

equations  is  found;  it  is  the  entry  corresponding  to  the  argu- 
ment Ja.   Thus  hj_  and  7i3  I  =  —  j  are  found,  and  hence  LJo 

—        — 


and  L3/(j  and  the  entire  range 


of  the  middle  class, 


in  terms  of  o,  is  known.  Call  the  range  in  absolute  units  Z. 
Then  l=L3  +  ^  and  I/  a  is  known  and  for  a  second  series  I  /a' 
can  be  similarly  determined.  Hence  a  /a'  ',  the  ratio  of  the 
variabilities  of  the  two  series,  is  determined. 

Again,  since  LJa  and  —  -  *  are  known,  Ll/(L3  +  L1)  is 

known,  and  this  gives  us  the  ratio  in  which  the  mean  divides 
the  true  range  of  the  central  class.  (Pearson  and  Lee,  1900.) 
The  foregoing  method  may  sometimes  be  advantageously 
employed  where  the  data  are  quantitative.  In  this  case 
the  numerical  value  of  I  is  known.  (Macdonell,  1902.) 

Consequently      /it  +  h2  =  — 
=    *        3 


is     known     and     hence 


"i 


,  the  standard  deviation,  is  found.     Since  Lv 


the  distance  of  the  mean  from  the  left-hand  boundary  of  n2f 
the  position  of  the  mean  is  known. 
The  probable  error  of  a  is 


3 


where 


and 


COKRELATED    VARIABILITY. 


49 


The   values  of    the   last   two   equations   may   be   obtained 
directly  from  Table  III. 

The  probable  error  of  Llt  or  rf  the  mean,  is 


=  .67449  - 


_  /    Ea   V    v  2  _ 
-  '     A" 


_n3(n-n3) 


THE  CORRELATION  OF  NON-QUANTITATIVE  QUALITIES. 

Pearson  (1900C)  has  ingeniously  discovered  a  method  of  ex- 
pressing correlation  quantitatively  when  the  variables  cannot 
be  so  expressed,  as,  for  example,  in  the  case  of  effectiveness 
of  vaccination.  Strictly,  this  method  assumes  normal  vari- 
ation in  variables,  but  it  can  be  employed  generally,  in 
default  of  a  better  method,  with  fairly  accurate  results. 

The  prime  requisite  is  that  the  qualities  to  be  compared 
shall  be  separable  into  two  grades,  an  upper  and  a  lower. 
For  example,  in  the  case  of  the  result  of  vaccination:  on 
the  one  hand,  either  presence  or  absence  of  a  scar;  on 
the  other,  either  recovery  or  death.  As  either  of  the 
second  pair  may  occur  with  either  of  the  first  pair,  four 
classes,  a,  6,  c,  d,  will  be  formed  altogether  and  a  correlation 
surface  like  the  following  may  be  made: 


a 

b 

a  +  b 

c 

d 

c  +  d 

a  +  c 

b  +  d 

n 

The  axes  y,—y  and  x,— x  probably  do  not  coincide  with  the 
axes  y  and  x  passing  through  the  "origin"  of  the  correlation 


50  STATISTICAL   METHODS. 

surface,  but  may  be  regarded  as  situated  from  those  axes  at 
the  respective  distances  h  and  k.  These  values  may  be 
found  from  the  formulae 


(a+b)_(c  +  rf)         /2M 
n  *    TT  •/  o 

a,  b,  c,  and  d  being  known,  h  and  fc  are  found  from  Table  IV. 
Then 


of  which  the  values  may  be  looked  up  in  Table  III,  or,  better, 
their  product  may  be  calculated  by  logarithms  as  follows: 


Find  also  log  hk,  h2,  and  k2.      To  find  r  solve  the  following 
equation  to  as  many  terms  as  may  be  necessary: 


4  -  6/i2  +  3)  (k*  -  G/c2  +  3)r5 


This  gives  us  a  numerical  equation  of  the  nth  degree  which 
can  be  solved  by  ordinary  algebraic  methods,  using  Sturm's 
functions  and  Horner's  method.  Or  it  can  be  solved  by 
successive  approximations  as  follows:  The  first  approxima- 
tion is  made  by  neglecting  all  powers  of  r  above  the  second 
and  solving  the  quadratic  (remembering,  that  if  ax2  +  bx  +  c  ==  0, 


CORRELATED   VARIABILITY.  51 

— 2^ —  "  ) »  anc*  taking  the  positive  root.  Substi- 
tute this  value  in  the  whole  equation  to  the  4th  power  for 
/(r),  and  in  the  first  derivative  of  the  same  equation  for  /'(r) 
(remembering  that  the  first  derivative  of  /(#)  is  obtained  by 
multiplying  each  term  in  f(x)  by  the  exponent  of  x  in  that 
term  and  diminishing  the  exponent  of  x  by  1).  The  correc- 
tion ^7-7^7  should  be  added  to  the  value  of  r  used  in  substi- 
tuting. Repeat  this  process  as  often  as  the  correction  affects 
the  fourth  place  of  decimals,  and  go  to  r5  if  necessary. 
The  probable  error  of  r  as  thus  determined  is 

found  as  follows:   First  calculate  the  relations  &=      _= 

k  —  rh 
and  /?2  =    ".    — .     Also  find 


and 


from  Table  TV.    Moreover, 

1 


Then, 

Prob.  error  of  r=-.  i[j(o 

(a  +  b)(d  +  c)  &*  +  2(ad  -  be)  fa 


which  can  be  easily  solved  by  substitution.  In  using  the 
foregoing  formula,  it  must  be  noted  that  "a  is  the  quadrant 
in  which  the  mean  falls,  so  that  h  and  k  are  both  positive." 
In  other  words,  a  +  c  >  b  +  d  and  a  +  6  >  c  +  d.  (Pearson,  '00°.) 

Example.  The  eye-colors  of  a  certain  set  of  people  (see  Bio- 
metrika,  II,  2  pp.  237-240)  and  of  their  great  -grandparents  were 
found  to  be  distributed  as  follows 


STATISTICAL   METHODS. 
Offspring. 


1 

2 

3 

4 

5 

6 

7 

8 

6 

C 
3 

| 

J^ 

o 

Q. 

« 

pq 

Q 

<B 

ofj 

pq 

j 

s 

pq 

m 

"S 

1 

§« 

^l 

|w 

!> 

g 

1 

1 

1 

1 

3 

« 

O 

(5 

3 

pq 

P 

S 

H 

•a 

1  .  Light  blue  

4 

3 

8 

5 

1 

21 

§ 

2    Blue—  dark  blue  .  . 

8 

177 

95 

76 

'  's 

39 

'si' 

"17" 

448 

J? 

3.  Gray  —  blue-green. 
4.   Dark  gray  —  hazel. 

1 
6 

69 
30 

85 
21 

52 
27 

2 
2 

20 

7 

26 
15 

1 
1 

256 
109 

5    Light  brown 

4 

4 

»H 

6    Brown                .    .  . 

2 

37 

27 

17 

3 

30 

20 

4 

140 

O 

7.  Dark  brown  

15 

20 

24 

3 

4 

9 

9 

84 

8    Black     

10 

13 

12 

2 

2 

7 

5 

51 

Totals  

21 

345 

269 

213 

17 

103 

108 

37 

1113 

It  was  desired  to  determine  the  correlation  between  the  eye-color 
of  the  offspring  and  that  of  their  great-grandparents.  Clearly  the 
ranges  of  the  classes  given  above  are  not  quantitatively  equal  nor 
determinable  Consequently  a  fourfold  table  was  formed  by  dividing 
the  population  into  those  having  eyes  whose  color  was  gray  blue-green, 
or  lighter,  and  those  having  dark  gray,  hazel,  or  darker  eyes.  This 
gives  a  good  basis  for  calculation  If  the  dark  gray  and  hazel  eyes 
had  been  grouped  with  the  lighter  eyes  it  would  have  made  quadrant 
a  entirely  too  large;  and  there  is  nothing  in  the  nature  of  the  data 
that  strongly  favors  one  division  more  than  another. 


7  25  -388 
1113 


.302785 


From  the  tables: 


Offspring 


1-3 

4-8 

Totals. 

1-3 

450 

275 

725 

4-8 

185 

203 

388 

Totals. 

635 

478 

1113 

Ol 

h 

.31 

.30 

.  39886 
.38532 

.01 

.01354 

«2 

k 

.15 
.14 

.18912 
.  17637 

.01 

.01275 

h  =  .38532  +  ( 1 .354  X  .002785)  =  .389091 
k  =  .17637  +  (1.275  X  .001060)  =  .177722 


CORRELATED   VARIABILITY. 


53 


Log    £=9.5900512 
Log    k  =  9.2497412 

Log  A2  =  9.1801024 
Log  k2  =  8.  499  4824 

Jthlf—  0-34  *7*                       h 

A2  =  .  151392 
A;2  =  .031585 

*  +  k2  .182t77 

4-A-2 
HQI^CQ 

Log  ta  -8.8397924 

h1f.=  ORQI.^n 

Log  (450X203  -27  5X185)  =4.607  1869 

Log  HK  =  -  log  2  rr  -  .091489  log  e 

=  9.2018201  -M8.9613689  +  9.63778428] 
=  9.2018201-0.0397332  =  9.1620869 


=  4.6071869-(9.1620869  +  2  log  1113)  =9.3521096 


.224962  =r 

Solving  .034575r2+r-.  224962  =  0, 

1  ±  X/f+4T034575  X  .224962) 
-2(7634575)— 


2_  !  =  _  .848608 

Coeff.  r4  = 


A:2-  1  =  -  .968415  CoeflF. 

+  -069150  X  2.848608  X  2.968415 


.136967 


.024363r4  +  .  136967r3  +  .03457  5r2  +  r  -  .224962  =  0. 
Applying  Newton's  approximation,  we  reach  the  result 

r  =  .2217. 
(7  5095  +  303530&>2  +  281  300^t 


Log  w0=log*ff-*log(l-r2) 

-Iog(l-r2)-log2] 

ht  +  k*-2rhk  =  0.152315,  l-r2  =  0.950850. 

Logw0=9.20182-9.989056-M9.637784  +  9.18274-9.978112-0.30103] 
=  9.1779797 


Log 


9.828975-4.569743-9.177980  =  4.081253. 


/?!=  0.358614 
Table  IV: 


.358        .13983 
22.2 
.4 

fa  =  .14006 


Log  E  .r  =  4  .  08  1  2530  +  \  log  7  4426  .  858 
E.r  =  0.03289 


#2=0.093794 


.093 


.03705 
27.3 
3.5 


4>2  =  .03736 


54  STATISTICAL   METHODS. 


QUICK  METHODS  OF  ROUGHLY  DETERMINING  THE  COEFFI- 
CIENT OF  CORRELATION. 
• 
The  method  just  described  may  be  used  in  lieu  of  the  rela- 

2x  11 
tion  r= — —    whenever  the  distributions  of  frequencies   of 

na1a2 

the  two  correlated  organs  are  normal.  An  exceedingly  sim- 
ple relation  that  is  independent  of  the  assumption  of  a  normal 
distribution  has  been  given  by 'Yule  ('00b)  as 

_^  ad  —  be 
T*~~  ad  +  bc' 

and  this  may  be  used  as  a  rough  approximation  to  the  coeffi- 
cient of  correlation. 

But  Pearson  ('00C)  has  shown  that  this  simple  relation  is 
not  nearly  as  close  to  the  true  r  as  the  following: 


where 

4abcd  .  \ 


The  superiority  of  the  value  rs  as  an  approximation  to  r^ 
justifies  the  additional  work  its  determination  demands. 

SPURIOUS  CORRELATION  IN  INDICES. 

When  two  characters  a  and  b  are  measured  in  each  indi- 
vidual of  a  series  of  individuals,  and  each  absolute  magnitude 
is  transformed  into  an  index  by  dividing  it  by  the  magnitude 
of  a  third  character  c  as  found  in  the  same  individuals,  a 
spurious  correlation  will  be  found  to  exist  between  the  indices 

of  -  and  -  (Pearson,  '97). 
c  c 

Let  Cj=the  coefficient  of  variability  of  a; 

C_      f  f  <  (  (  (  (I  I  (       -L. 

2=  b; 

C__<<  t f         t (          1 1          ( (    . 
3~~  C, 

rQ=  "  "  spurious  correlation. 

C  2 


COERELATED    VARIABILITY.  55 

The  precise  method  of  using  r0  in  modifying  any  determi- 
nation of  r  is  uncertain.  Pearson  recommends  using  r  —  r0 
as  the  true  measure  of  "organic  correlation"  in  the  case  of 
indices. 

HEREDITY. 

Heredity  is  a  certain  degree  of  correlation  between  the 
abmodality  of  parent  and  offspring.  The  statistical  laws  of 
heredity  deal  not  with  relations  between  one  descendant  and 
its  parent  or  parents,  but  only  with  mean  progeny  of 
parents.  Any  group  of  selected  parents  is  called  a  parentage, 
the  progeny  of  a  parentage  is  called  a  fraternity. 

Three  categories  of  inheritance  have  long  been  recognized 
(Galton,  1888,  p.  12).  These  are:  (1)  blending  heritage  illus- 
trated by  stature  in  man;  (2)  alternative  heritage,  illustrated 
by  human  eye-color;  and  (3)  mixed  heritage,  illustrated  by 
the  piebald  condition  of  the  progeny  of  mice  of  different 
colors.  The  immediately  following  statistical  laws  of  inherit- 
ance hold  especially  for  blending  heritage. 

In  u  11  i parental  inheritance,  as  in  budding  or  asexual 
generation,  heredity  of  any  character  is  measured  by  the  coef- 
ficient of  correlation  between  the  abmodality  in  a  parentage 
and  the  abmodality  of  the  corresponding  fraternity.  More 
strictly,  since  the  variability  of  the  character  in  the  second 
generation,  a2,  may  (as  a  result  of  selection  or  of  environ- 
mental change)  be  different  from  the  variability  of  the  char- 
acter in  the  first  generation,  <71?  the  index  should  be  taken  as 

r— ,  called  the  coefficient  of  regression. 

The  probable   error  of  this  determination  is 

.6745^.  /I  -r122    . 

—  I/ — ,  in  which  r12  means  the  correlation  coem- 

cient  between  the  filial  character  and  that  of  the  single  parent 
under  consideration. 

The  variability  of  the  fraternity  is  to  variability  of  offspring 
in  general  as\/l—r2  is  to  1. 

In  bipareiital  inheritance,  if  there  is  no  evidence  of 
assortative  mating,  or  correlation  between  the  two  parents  in 
the  character  in  question,  the  mean  abmodality  of  any  f rater- 


56  STATISTICAL   METHODS. 

nity  will  be 


where  7^=  average  abmodality  of  fraternity; 

h2=  average  abmodality  of  male  parent; 

h3=  average  abmodality  of  female  parent; 

r2=  correlation     coefficient    between    fraternity    and 

female  parent; 
r3=  correlation  coefficient  between  fraternity  and  male 

parent; 

o1  =  standard  deviation  of  fraternity; 
o2=  standard  deviation  of  male  parent; 
<73=  standard  deviation  of  female  parent. 
When  assortative  mating  occurs,  as  is  usually  the  case,  the 
abmodality  of  a  fraternity  is  given  by 


where   rx=  correlation    between    male    and   female    parents. 
The  other  letters  have  the  same  signification  as  before. 

The  strength  of  heredity  in  assortative  mating  is  measured 
by  the  formula 


To  find  the  coefficient  of  correlation  between 
brethren  from  the  means  of  the  arrays. 

This  is  given  by  the  formula 


where  rij  is  the  number  of  the  brethren  in  an  array  [and  there- 
fore JrijCnj  —  1)  is  the  number  of  possible  pairs  of  brothers  in 
that  array];  Al  is  the  mean  value  of  the  array;  a  is  the 
standard  deviation  of  the  character  in  the  brethren  taken 
all  together,  n  is  the  total  number  of  variates,  and  A2  is  the 
average  of  the  brethren.  This  method  will  be  found  useful 
where  to  take  all  possible  pairs  of  brethren  would  be  found 
a  work  of  too  great  magnitude  (Pearson,  Lee,  etc.,'99,  p.  271). 


CORRELATED    VARIABILITY.  57 

Galtou  ('97)  has  shown  that  an  individual  inherits  not  only 
from  his  parents,  but  also  from  his  grandparents,  great-grand- 
parents, and  so  on.  The  heritage  from  his  2  parents  together 
is,  on  the  average,  50$  or  £  of  the  whole  ;  from  the  4  grand- 
parents 25#  or  J  ;  from  the  8  great-grandparents  12.  5#  or  £  ; 

from  the  ?ah  ancestral  generation  —  of  the  whole  ;  the  total 

heritage  adding  up  100$.     This  law  has  been  generalized  by 
Pearson  ('98)  as  'follows  : 


where  /i!   =  average  abmodality  of  fraternity. 
CTO  =  standard  deviation  of  fraternity* 
cTj  ,  o-a  .  .  .  crs  =  standard  deviation  of  mid-parent  of 
1st,  2d  ...  sth  ancestral  generation. 
ki  =  abmodality  of  mid-parent  of  1st  ancestral  genera- 

tion. 
&2,  #3  .  .  .  Jcs  =  abmodality  of  mid-parent  of  2d,   3d 

.  .  .  sth  ancestral  generation. 

The  abrnodality  of  the  mid-parent  of  any  degree  of  ancestry 
may  be  taken  as  the  average  abmodality  of  all  the  contributory 
ancestors  of  that  generation. 

MENDEL'S  LAW  OF  ALTERNATIVE  INHERITANCE. 

In  1865  Gregor  Merdel  published  an  account  of  his  experi- 
ments in  Plant  Hybridization  and  reached  the  following  laws, 
which  have  been  abundantly  confirmed  in  certain  experi- 
ments. 

First  Case.  The  two  parents  differ  in  one  character  (the 
antagonistic  peculiarity)  —  case  of  monohybrids. 

Of  the  two  antagonistic  peculiarities  the  cross  exhibits 
only  one;  and  it  exhibits  it  completely,  so  as  not  to  be  dis- 
tinguishable in  this  regard  from  one  of  the  parents.  Inter- 
mediate conditions  do  not  occur  [in  alternative  heritage]. 

2.  In  the  formation  of  the  pollen  and  the  egg-cell  the  two 
antagonistic  peculiarities  are  segregated;  so  that  each  ripe 
germ-cell  carries  only  one  of  these  peculiarities. 


58  STATISTICAL  METHODS. 

Of  the  two  antagonistic  peculiarities  united  in  the  cross, 
that  which  becomes  visible  in  the  soma  is  called  by  Mendel  the 
dominating,  that  which  lies  latent  is  called  the  recessive  char- 
acter. What  determines  which  character  shall  be  dominating 
is  still  unknown,  and  the  determination  of  this  point  offers  an 
enticing  field  of  inquiry.  In  some  cases  the  dominating  form 
is  the  systematically  higher,  in  others  it  is  the  older  or  ances- 
tral form. 

The  law  of  dichotomy  may  now  be  developed.  When  a 
mongrel  (monohybrid)  fertilization  takes  place  the  zygote  con- 
tains both  the  dominant  quality  (abbreviated  d)  and  the  re- 
cessive quality  (r).  In  the  early  cleavages  d  and  r  are  both 
passed  over  into  both  the  daughter-cells;  but  apparently,  at 
the  time  of  segregation  of  the  germ-cells,  the  somatic  cells 
are  provided  with  d  only,  while  the  germ-cells  retain  both 
qualities.  In  the  ripening  of  these  germ-cells,  probably  in 
the' reduction  division,  d  and  r  come  to  reside  in  distinct  cells, 
so  that  we  have 

of  the  female  cells  50%d  +  50%r,  and 
of  the  male  cells     50%d  +  50%r. 

If  now  mongrels  are  crossed  haphazard,  a  male  d  cell  may 
unite  with  either  a  female  d  cell  or  with  a  female  r  cell;  like- 
wise a  male  r  cell  may  unite  with  a  female  d  or  a  female  r  cell. 
Consequently  in  the  long  run  we  shall  have  of  all  the  zygotes 

25%d,  d  +  50%d,  r  +  25%r,  r, 

or  50%  of  the  zygotes  hybrid  and  50%  of  pure  blood,  and  of 
the  latter  half  exclusively  maternal  and  half  paternal.  But 
since  the  soma  developed  from  the  hybrid  germ-cell  has  the 
dominant  character,  we  shall  have 

75%  of  the  cases  with  the  dominant  character; 
25%  "    "       "        "      "    recessive 

and  this  agrees  with  various  empirical  results,  of  which  the 
following  from  Correns  is  instructive.  A  cross  was  obtained 
between  a  variety  of  pea  with  a  green  (</)  germ  and  one  having 
a  yellow  (y)  germ.  Yellow  is  dominating. 


CORRELATED    VARIABILITY. 


59 


Gen.  1. 

31  y  (hybrid)  peas  produced  12  plants* 
these  bore: 

Gen.  2. 

775  y  (hybrid  +  y)  peas  (  =75.8%)              247  g  (pure-blooded) 
21  plants  were  produced:                          peas  (  =24.2%). 

7  (33%)  pure-      14  (66%)  hybrids, 
blooded  y,           because  they                        20  plan 
because  they               bore  : 
bore:                           | 

ts  bore  : 

Gen.  3. 

292  y  peas'         462  y                    149  g              670  green  peas, 
(hybrid  +  y)         (pure-blooded) 
peas  (=76.4%)    peas  (=23.6%) 

It  is  clear  that  if  this  process  of  crossing  of  the  hybrids 
continues,  the  proportion  of  hybrids  to  the  whole  population 
will  diminish;  for  the  share  of  pure-blooded  forms  breeds 
true;  while  the  originally  equal  share  of  hybrids  is  repeatedly 
halved. 

If  the  hybrid  is  crossed  with  one  of  the  parents  instead  of 
with  another  hybrid,  we  will  get 

(1)  (d  +  r)d=d,  d+d,  r,      and 

(2)  (d  +  r)r=d,  r+r,r. 

In  (1)  all  of  the  progeny  will  appear  of  the  dominant  type. 
In  (2)  one-half  will  appear  of  that  type.  This  again  agrees 
with  experiment. 

Second  Case.  The  two  parents  differ  in  respect  to  two 
characters — case  of  dihybrids.  Imagine  a  lot  of  ripe  germ- 
cells  with  the  antagonistic  qualities  of  any  pair  separated 
according  to  the  second  principle  stated  at  the  outset.  A 
indicates  the  one  pair  of  qualities  and  B  the  other;  then  we 
shall  have  nine  classes  of  zygotes,  the  proportion  of  each  of 
which  is  as  follows: 

A. 


B.     6.25%d,  d;  12.5%d,  r;  6.25%r,  r. 
A.  50%d,  r 


B.     12.5%d,d;  25%d,  r;  12.5%r,  r. 

A.  25%r,  r 


B.     6.25%d,  d;  12.5%d,  r;  6.25%r,  r. 


60  STATISTICAL   METHODS. 

Thus  the  first  class  has  6.25%  purely  dominant  in  both 
characters;  the  second  class,  12.5%  purely  dominant  in  one 
character  and  hybrid  in  the  other,  and  so  on.  Recalling  that 
hybrid  zygotes  produce  somas  with  the  dominant  character, 
it  follows  that  the  progeny  appear  as  follows: 

Ratios 
A.  dom.  +  B.  rec  ...........   18.75%         3 

'A.  rec.    +B.  dom  ..........    18.75%         3 

A.  dom.  +  5.  dom  ..........   56.25%         9 

A.  rec.    +B.  rec  ...........     6.25%         1 

This  result  again  agrees  with  experiment.  The  resulting 
mixture  of  characters  in  tri-  to  poly  hybrids  may  be  likewise 
predicted,  by  extending  the  principles  already  laid  down. 

MEASURE  OF  DISSYMMETRY  IN  ORGANISMS. 

A  Dissymmetry-Index,  3,  measuring  the  average  de- 
gree of  asymmetry  in  the  right  and  left  organs  of  bilateral 
organisms,  has  been  proposed  by  Duncker  (1903). 

First  a  series  of  integral  differences  —3,  —2,  —1,  0,  1,  2, 
3,  4,  etc.,  between  the  right-  and  left-side  measurements  of 
the  organ  in  question  is  made,  and  the  frequencies  of  each 
integral  difference  (reckoning  to  the  nearest  integer)  is  found. 
The  average  of  the  difference  series  is  the  difference  of  the 
averages  of  the  right-  and  left-side  measurements,  and  the 
standard  deviation  of  the  difference  is  given  by 


in  which  the  subscripts,  refer  to  the  bilateral  series  of  which 
the  asymmetry  is  to  be  found,  and  r  is  the  coefficient  of  cor- 
relation between  the  two  sides. 

Let  d'  represent  any  positive  differences  in  the  series,  and 
d"  any  negative  differences;  and  let  /,/,  //,  etc.,  represent 
the  frequencies  of  the  negative-difference,  classes,  and  //', 
//',  etc.,  the  frequencies  of  the  positive-difference  classes. 
Then  the  asymmetry-index 


_ 


CORRELATED    VARIABILITY.  61 

Example.  Absolute  difference  between  dextral  (d)  and 
sinistral  (s)  lateral  edges  (L)  of  carapace  of  right-handed 
fiddler-crabs — Gelasimus  pugilator  (Yerkes,  1901;  Duncker, 
1903)' 

d=Ld-Ls:   -10        1        23 
/:       1     63     310     23     3 

310X  1  +  23X2  +  3x3  =  365,     ^(/0  =  336. 
l,     2<j")  =  \,     n=400. 

336X365-1X1_  122639 
400X366   "    ~  146400 


62  STATISTICAL   METHODS. 


CHAPTER  V. 

SOME  RESULTS  OF  STATISTICAL  BIOLOGICAL  STUDY. 

It  is  hoped  that  the  following  analysis  of  the  literature, 
although  not  complete,  will  prove  suggestive  and  otherwise 
useful.  Numerical  results  are  occasionally  given.  These  are 
intended  to  be  used  in  making  comparisons  with  numerical 
results  obtained  in  the  same  field  and  thus  to  assist  in  the 
interpretation  of  such  results.  The  literature  references  are 
to  the  Bibliography  which  follows  this  chapter,  in  which  the 
titles  are  arranged  by  author  and  date. 

GENERAL. 

EXPOSITIONS,  ADDRESSES,  ETC.:  Amann,  '96;  Ammon,  '99; 
Camerano,  '00b,  '01,  '02;  Davenport,  '00,  '00d,  Olb; 
Duncker,  '99b;  Eigenmann,  '96;  Galton,  '01;  Gallardo, 
'00,  '01,  'Olb;  Ludwig,  '00,  '03;  Redeke,  '00;  Volterre, 
'01. 

TEXT-BOOKS:  Galton,  '89;  Bateson,  '94;  Duncker,  '00; 
Pearson,  '00;  Vernon,  '03. 

METHOD:  Camerano,  '00;  Engberg,  '03;  Fechner,  '97; 
Galton,  '89,  '02;  Heincke,  '97;  Johannsen,  '03;  Pear- 
son, '94,  '95,  '96,  '97,  '97b,  '98,  '00°,  'Old,  '02C,  '02f,  '02e, 
'02m,  '02n,  '03e;  Pearson  and  Lee,  '00;  Sheppard,  '98, 
'98b,  '03;  Verschaffelt,  '95;  Wasteels,  '99,  '00;  Yule, 
'97,  '97b,  '00,  '00b,  '03. 

VARIABILITY. 
General. 

Frequency  polygon,  its  significance;  its  dependence  on 
time,  place,  and  conditions:  Burkill,  '95;  Kellerman,  '01; 
Tower,  '02;  Shull,  '02;  Yule,  '02;  Johannsen,  '03. 

Proper  value  of  ratio  of  first  to  second  prizes:  Galton,  '02; 
Pearson,  '02k. 


STATISTICAL   BIOLOGICAL   STUDY.  63 

Coefficient  of  variability;  significance:  Pearson, '96;  Brew- 
ster,  '97;  Duncker,  '00b;  Davenport,  '00e. 

Mutations:  Bateson,  '94;  Howe,  '98;  deVries,  '01-' 03; 
Weldon,  '02°. 

Individual  vs.  specific  variation:  Brewster,  '97,  '99;  Field, 
'98;  Mayer,  '02;  Davenport  '03*>. 

Variability  independent  of  sexual  reproduction:  Warren, 
;99,  '02;  Pearson  and  others,  '01  c,  pp.  359-362. 

Relative  variability  of  the  sexes: — in  man,  Pearson,  '97C; 
Brewster,  '99;  Pearl,  '03;  in  crabs,  Schuster,  '03. 

Relative  variability  of  primitive  and  modern  races: — in  man, 
primitive  races  less  variable:  Pearson,  '96,  p.  281;    Pearson 
(and  others),  '01 c,  p.  362. 
Man. 

Stature. — Seriation  for  adults  of  different  races:  Bavari- 
ans, Ammon,  '99;  United  States,  recruits,  Baxter,  '75,  Pear- 
son, '95,  p.  385;  various,  Macdonell,  '02;  English  middle 
upper  classes,  Galton,  '89,  Pearson,  '96,  p.  270;  Germans, 
Pearson,  '96,  p.  278;  French,  Pearson,  '96,  p.  281;  Cam- 
bridge University  students,  Pearson,  '99. 


Lot. 

n 

A 

0 

C 

Engl.  upper  middle  class  1 

683 

69.215"  ±.  066 

2.592"  ±.047 

do.         husbands  . 

200 

69.  135"  ±.126 

2.628"  ±.  089 

3.66 

Cambridge  Univ.  students 

68.  863"  ±.054 

2.522"  ±.048 

cm. 

cm. 

English  fathers  

1078 

171.95 

6.81 

3.99 

English  sons.     .                   . 

1078 

174.40 

6.94 

3.98 

U.  S.  recruits  

25878 

170.94 

6.56 

3.84 

N.  S.  Wales,  criminals..  .  . 

2862 

169.88 

6.58 

3.80 

Frenchmen     .  .                    . 

284 

166.80 

6.47 

3.88 

English  criminals       

3000 

166  .  46 

6.45 

3.88 

French,  Lyons  

166.  26  ±.53 

5.50±  .37 

Germans 

390 

156.93 

6.68 

4.02 

in. 

in. 

Engl  .  upper  middle  class  ? 

652 

64.043  ±.061 

2.325  ±.043 

do.           wives.  .  .  . 

200 

63.869±.110 

2.  303  ±.078 

Cambridge  Un.  students  ? 

63.  883  ±.130 

2.361  ±.092 

3.69 

French,  Lyons  ?  .  . 

154.02  cm.  ±.52 

5     5.45  ±.37 

Seriation  at  different  ages:  British  infant  at  birth,  Pearson, 
'99;  school  children,  Bowditch,  '91;  St.  Louis  schoolgirls, 
Porter,  '94,  Pearson,  '95,  p.  386;  Australian  adult  whites, 
Powys,  '01. 


64 


STATISTICAL    METHODS. 


Lot.                          Average.  a 

New-born  infant,  British  $ .  20 . 503 ±  . 028  in.  1 . 332 ±  . 020 

?.   20.124±  .025  "  1.117  ±.018 

St.  Louis  schoolgirls 118.271  cm.  2.776 

Australian  whites: 


Age, 

Years 

20-25 
25-30 
30-40 
40-50 
50-60 


Average. 


66.95 
67.30 
67.15 
66.91 
66.74 


60  &  over  66.26 


62.50 
62.76 
62.44 
62.96 
62.22 
61.31 


2.475 

2.562 
2.587 
2.618 
2.633 


2.365 
2.432 
2.303 
2.555 

2.591 
2.300 


$ 

3.70 
3.81 
3.86 
3.91 
3.95 
4.04 


C 

6.500 
5.849 


3.79 
3.87 
3.69 
4.06 
4.16 
3.75 


Weight. — Seriations  at  different  ages,  British:  Infants, 
Pearson,  '99;  University  students,  Pearson,  '99;  5552  Eng- 
lishmen, Sheppard,  '98. 

Lot.  Average.  o  C 

New-born  infants,  $ 7  .301  ±  .0241b.        1.144±.017  15.66% 

? 7.073±.021  1.006±.015  14.23 

Cambridge  Univ.  students,  $  152. 783  ±  .35  16. 547  ±.25  10.83 

?125.605±.77  14.030±.57  11.17 

Skull. — Cephalic  index:  Bavarians,  Ranke,  '83;  6800  20- 
year  old  Badeners,  working  class,  Ammon,  '99,  p.  85;  various 
races,  Pearson,  '96,  p.  230,  Macdonell,  '02. 

Lot.  n 

Bavarian  peasants 100 

Baden  recruits 6748 

Modern  Parisians 

French  peasants 53 

Cambridge  students 1000 

Criminals  (British) 100 

Brahmans  of  Bengal 100 

Whitechapel  English.  .  , 107 

Maquada  race 

Skull  capacity:  coefficients  of  variability.  Fawcett  and 
Lee,  '02. 

Lot.  $ 

Andamanese 5 . 04 

Ainos 6.89 

Negroes 7.07 

Low-caste  Pun  jabs  .  .  7.24 

Parisian  French 7  . 36 

Kanakas 7 . 37 

17th  Century  English.  7 . 68 


A 

a 

C 

83.41 

3.58 

4.29 

81.15 

3.63 

4.48 

79.82 

3.79 

4.74 

79.79 

3.84 

4.81 

78.33 

2.90 

3.70 

76.86 

3.65 

4.75 

75.77 

3.37 

4.44 

74.73 

3.31 

4.43 

72.94 

2.98 

3.95 

? 

Lot. 

$ 

9 

5.59 

Naquadas  ...    . 

7  72 

6.92 

6.82 

Germans  

.   7.74 

8.19 

6.90 

Egyptian  mummies.  , 

.   8.13 

8.29 

8.99 

Polynesians  

.   8.20 

5.55 

7.10 

Italians  

8  34 

8.99 

6.68 

Modern  Egyptians.  . 

.   8.59 

7.17 

8.15 

Etruscans  .  . 

9.58 

8.54 

STATISTICAL   BIOLOGICAL    STUDY.  65 

Various  cranial  dimensions,  Lee  and  Pearson,  '01. 

Other  Organs. — Coefficient  of  variability  of  bones  of  skele- 
ton of  French  and  Naquada  (C.  of  limb-bones,  4.53-5.57), 
Warren,  '97;  appendicular  skeleton,  Pearson,  '96;  finger- 
bones,  Lewenz  and  Whiteley,  '02;  seriation  of  position  of 
spinal  nerves,  Bardeen  and  Elting,  '01;  various  organs  in 
diverse  races,  Brewster,  '97,  '99. 
Mammalia. 

Relative  variability  of  specific  and  generic  characters  in 
various  mammals  the  former  being  greater,  Brewster,  '97; 
seriation  of  number  of  Miillerian  glands  in  Sus  scrofa,  n,  2000; 
A,  3.501  ±.025;  a,  1.6SO±.018;  C,  48.0,  Davenport  and  Bui- 
lard,  '96. 
Aves. 

Seriations  of  various  proportions  of  N.  A.  birds,  Allen,  '71; 
characters  of  Lanius  ("shrike")  and  its  races,  Strong,  '01; 

Lot.                                n               A  a  C 

Shrike,  length  L.  wing  $ 168  99.06mm.     2.74mm.    281 

'     ? 112  97.98  2.64  2*69 

tail  length 4 141  101.57  3.48  343 

? 95  99  55  3.63  3.65 

bill  length,  $ 164  12.01  0.71  589 

"         "         ? 112  11.71  0.63  5^35 

"    depth,  a 126  9.27  0.42  457 

? 85           8.95  0.41  4.61 

melanism  of  crown,  $ 144  83.57%  3.0%  358 

? 99  83.66  3.19  3.81 

'"    upper  tail-coverts  i    142  53.13  15.42  2902 

1      ?   104  47.98  18.99  39^58 

Curvature  of  cuimen 29.94°  2.74°  9. 15 

Eggs,  proportions:    Passer  domesticus,  Bumpus,  '97,  Pear- 
son, '02e;   various  species,  Latter,  '02. 

Av. 

Length,  Length,  mm.  Breadth,  mm. 

Species.  Bird,       n  A  a  C  A          a          C 


Cuckoo 

in. 
14 

243 

2° 

40 

1. 

059 

4. 

72 

16.54 

.650 

3  93 

Blackbird  

10 

114 

29. 

44 

1  . 

357 

4 

01 

21  73 

!787 

3  62 

Song-thrush  .... 
Starling  . 

9 

8-8.5 

151 
27 

27 
29 

,44 

78 

0 
1. 

999 
097 

3, 
3 

.04 
08 

20.69 
21  76 

.516 
.423 

2.  '50 
1  94 

Yellowhammer.  . 
Tree-pipit  
Meadow-pipet  .  . 
House-sparrow 
(English)  
House-sparrow 
(American)  .  .  . 
Hedge-sparrow.  . 
Robin  . 

7 
6.5 
6 

6 

6 
6 
6 

32 
27 

74 

687 

868 
26 
57 

21  . 
20 
19. 

21 

21 
20 
20 

55 

01 

72 

.82 

.32 
.12 
.22 

0. 
0. 

1. 
1. 

1 
0 

() 

682 
698 
250 

195 

05 

.810 
,857 

3. 

3 
6. 

5. 

4. 
4 
4 

17 
49 
37 

47 

92 
.02 
24 

16.04 
15.09 
14.56 

15.51 

15.34 
14.73 
15  43 

.405 
.449 
.561 

.525 

.415 
.477 

2'53 
2.97 
3.84 

3.38 

2.81 
3  09 

Linnet  .  .. 

5.5-6 

65 

17 

.14 

0, 

598 

3. 

.49 

13.33 

.358 

2^69 

66  STATISTICAL   METHODS. 

Amphibia. 

Seriations   of   variations  in   position    of   pelvic   girdle   in 
Necturus,  Bumpus,  '97. 
Pisces. 

Geographical  races:  in  Leuciscus,  Eigenmann,  '95;  in 
adjacent  lakes,  Moenkhatis,  '96;  in  schools  of  herring,  Heincke, 
'97;  in  flounders,  Bumpus,  '98;  in  mackerel,  Williamson,  '00. 
See  under  Local  Races. 

Various  species:  Pimephales  fin-rays  and  scales  of  lateral 
line,  Voris,  '99;    Zeus  faber,  an  ancestral  Pleuronectid,  has 
its  plates   symmetrical    in   only  23.6%    of   the   individuals, 
Byrne,  '02;    dimensions  of  141  PetromyzorT,  Lonnberg,  '93. 
Tracheata. 

Lepidoptera. — Seriations  of  wing  dimensions  of  Thyreus 
abbotti.  Field,  '98;  number  of  "eye-spots"  on  wing  of  Epi- 
nephele,  Bachmetjew,  '03;  number  of  spots  on  different 
species  of  the  genus  Papilio,  Mayer,  '02;  breadth  of  wing, 
985  Strenia  clathrata  C=4.57,  Warren,  '02. 

Aphidce. — Asexually  produced  offspring  show  an  average 
variability  of  60%  that  of  the  race,  Warren,  '02,  p.  144; 
seriation  of  fertility,  empirical  mode  =7  young,  Warren,  '02, 
p.  133;  reduced  variability  of  the  earlier  generations,  because 
they  include  only  such  as  can  produce  fertile  offspring,War- 
ren,  '02. 

Dimension.  Grandmothers.  Children. 

a  C  a  C 

Frontal  breadth 2.28mm.    6.07%         2.96mm.       8.26 

Length  R.  antenna 7.36  8.77  10.94  12.97 

Lengthantenna 
Frontal  breadth 

Myriapoda, — Lithobius:     seriations   of    length   of    adults, 
C,  for  4 's=  10.97;   ?'s=  11.25;  number  of   prosternal  teeth; 
of  antennal  joints;   of  coxal  pores  in  which  C  varies  from  9.9 
to  15.4,  Williams,  '03. 
Crustacea. 

Podophthalmata. — Seriations  of  12  dimensions  of  right- 
handed  and  left-handed  lt  fiddler-crabs,"  Gelasimus  pugilator, 
C  varies  from  7.0  to  11.1,  Yerkes,  '01;  relative  variability  of 
male  and  female  Eupagurus  prideauxi  from  deep  and  from 
shallow  water.  Schuster,  '03;  forehead  breadths  of  Carcinus 


STATISTICAL   BIOLOGICAL   STUDY.  67 

moenas,  Weldon,  '93,  Pearson,  '94;  various  dimensions,  Cran- 
gon,  Weldon,  '90;  length  of  rostrum,  Palaemon  serratus, 
Thompson,  '94,  Pearson,  '94;  number  of  rostral  teeth  of 
PalaBinonetes,  Weldon,  '92b,  Pearson,  '95,  Duncker,  '00. 

Lot.  A,  mm.  n,  mm.  C,  % 

Eupagurus,  short  edge  of  R.  chela: 

$  deep  water 9. 708  ±.085  2.76  28.5 

$  shallow  water 10. 272 ± .075  2.59  25.2 

?  deep  water 7. 400  ±.033  1.06  14.3 

$  shallow  water 7.485±.029  1.02  13.6 

Eupagurus,  long  edge  of  R.  chela: 

$  deep  water 17. 97  ±.14  4.73  27.8 

$  shallow  water 18. 68+.  13  4.38  23.5 

?  deep  water 14.14±.06  1.67  11.9 

9  shallow  water 13. 97  ±.05  1.82  13.0 

Eupagurus,  carapace  length: 

$  deep  water 8. 59  ±.05  1.67  19.4 

$  shallow  water 7. 54  ±.03  0.94  12.5 

?  deep  water 7.12±.03  0.86  12.1 

Palaemonetes  vulgar  is,  dorsal  spines  .  8.28  0.81  9.83 

ventral  spines.  2.98  0.45  15.03 

Palaemonetes,  varians,  dorsal  spines  .  4.31  0.86  20.00 

ventral  spines.  1.70  0.48  28.26 

Amphipoda. — Seriations    of   lengths    of   body,    of   second 
antennse,  and  of   ratio  of   second   antenna?   to   body-length, 
Smallwood,  '03. 
Annelida. 

Chcetopoda. — Teeth  on  jaws  of  Nereis  virens.     Right:  A  = 
10.055 ±.045,  o=  1.339 ±.032,  (7=13.3%;    Left:  A  =  10.00 ± 
.044,  <7=1.306±.031,  (7=13.1%,  Hefferan,  '00. 
Brachiopocla. 

Seriation  of  widths  breadth,  width  of  sinus  -f-  depth,  num- 
ber of  plications  on  ventral  and  dorsal  valves  in  sinus  and  on 
fold,  Cummings  and  Mauck,  '02. 
Bryozoa. 

Number  of  spines  on  statoblasts  of  Pectinatella  magnifica. 
A  =  13.782±.031,  o=  1.318 ±.022,  C=9.57±.16,  Davenport, 
'00«. 
Moll  u  sea. 

Gastropoda. — Frequency  polygons  of  ventricosity,  weight, 
and  index  of  Littorina  littorea  for  3  British  and  10  American 
localities — greater  variability  in  America.  Index:  <7£=2.3%, 


68  STATISTICAL   METHODS. 


B=2.6%,  Cj  =  3.0%,  Bumpus,  '98,  Duncker,  '98; 
critical,  Bigelow  and  Rathbun,  '03;  seriations  of  length, 
ratio  of  diameter  to  length,  ratio  of  aperture  to  length, 
apical  angle,  number  of  whorls,  color  of  aperture  lip,  and 
depth  of  suture  between  whorls  in  Nassa,  Dimon,  '02;  seria- 
tions of  shell-index  and  spinosity  of  lo  in  different  parts  of  a 
river  system,  Adams,  '00;  variability  of  adult  Clausilia 
laminata  less  than  that  of  young,  15:13,  ascribed  to  periodic 
selection,  although  average  size  not  altered,  Weldon,  ''01; 
variability  of  bands  of  Helix  nemoralis  in  one  spot  of  America, 
Howe,  '98;  in  different  localities  near  Strasburg,  Hensgen,  '02. 
Lamellibranchiata.  —  Seriation  of  number  of  ribs  of  Car- 
dium,  Baker,  '03;  Pecten;  ray-frequency,  Lutz,  '00,  Daven- 
port, '00,  '03  a  '03b;  change  in  proportions  with  age,  acquisi- 
tion of  new  symmetry  about  transverse  axis;  definition  of 
form  units  from  different  localities,  Davenport,  '03,  '03b. 

Lot.  Number  of  Rays. 

Pecten  irradians'  A  a  C 

Cold  Spring  Har.,L.  I.,  R.  valve  17.  353  ±.018  0.876  ±.013  5.05  ±.07 

Cutchogue,  L.  I.,  R.  valve  ....    16.534  ±  .034  0.852  ±  .024  5.32  ±  .36 

Cold  Spring  Har.,  L.  valve  ...   16.790±.022  0.916±.015  5.46±.09 

Cutchogue,  L.  valve  .........   15.954±.105  0.881±.075  5.52±.49 

Pecten  opercularis: 

Eddystone,  R  valve  ........   17.478±.029  1.000±.020  5.72  ±.12 

Irish  Sea,  R.  valve  ..........    18.101  ±.029  1.074±.021  5.93±.ll 

Firth  of  Forth,  R.  valve  .....    17.673±.027  1.117±.019  6.32±.ll 

Pecten  gibbus- 

Tampa,  Fla~,R  valve  ........  20.512±.030  0.991±.021  4.83±.10 

Pecten  ventricosus: 

San  Diego,  Cal.,  R.  valve  ....   19.495  ±.087  0.885  ±.019  4.55  ±.10 

Ecliiiiodermata. 

Seriation  of  ray-frequency  in  starfish,  Crossaster  papposus: 
A  =  12.391,  C=0.788,  0=6.36%,  Ludwig,  '93b. 
Coeleiiterata. 

Scyphomedusce.  —  Seriation  of  number  of  tentaculocysts  of 
Aurelia  aurita:  n=3000,  empirical  range  4-15;  empirical 
mode=8,  genital  sacs,  M'=4,  range,  2-10,  Browne,  '95,  '01. 

Hydromedusce.  —  Seriation  of  number  of  radial  canals, 
gonads,  gastric  lobes,  and  tentacles  of  Gonionemus,  Hargitt, 
'01;  radial  canals  and  lips  of  Pseudoclytia  pentata,  Mayer,  '01, 
Davenport,  '02;  radial  canals,  etc.,  of  Eucope,  Agassiz  and 
Woodworth,  '96. 


STATISTICAL    BIOLOGICAL    STUDY.  69 

Lot.  A  a  C 

[ocyltia,  num.  radial  canals 5.004±  .094         0.441          8.81 

lips 4.868±.012         0.556          11.4 

Protista. 

Paramecium  recently  divided,  Simpson,  '02;  seriation  of 
diameter  of  Actinospherium  and  number  of  cysts  and  nuclei 
in  body,  Smith,  '03;  outer  and  inner  diameters  of  shell  of 
502  Arcella  vulgaris,  Pearl  and  Dunbar,  '03;  various  diatoms, 
Schroter  and  Vogler,  '01. 

Lot.                                  A  a  C 

Paramecium,  length  n 229.05  19. 15  8.36% 

breadth 68.13  9.16  13.44 

index 29.91  4.03 

Arcella,  outer  diameter 55.79±.17  5.73±.12  10.27±.22 

inner  diameter 15.91±.07  2.17±.05  13.66±.30 

Plants. 

GENERAL. — Multimodal  polygons  especially  frequent  in 
plants,  Ludwig,  '97;  critical,  Lee,  '02;  Pearson,  '02h. 

RAY-FLOWERS  IN  COMPOSITE. — Seriation  of  ray-frequency 
of  Coreopsis,  de  Vries,  '94;  of  Senecio  nemorensis,  S.  fuchsii, 
Cent  urea  cyanus,  C.  jacea,  Solidago  virga  aurea,  Achilla  mille- 
folium,  Ludwig,  '96;  ray-frequency  in  Chrysanthemum, 
Ludwig,  '97C,  Lucas,  '98,  Tower,  '02,  Pearson  and  Yule,  '02; 
Helianthus,  Wilcox,  '02;  Bellis  perennis,  Ludwig,  '9Sb;  Soli- 
dago  serotina,  Ludwig,  '00b;  Arnica  montana,  Ludwig,  '01; 
Aster,  Shull,  '02. 

Num.  Ray-flowers.  A  o  C 

Aster  shortii 14.000±.068  1.526±.048  10.90 

A.  novaj-anglise 42.874±.302  6.308±.213  14.71 

A.punicens 36.672±.107  4.480±.076  12.22 

A.  prenanthoides 28.080±  .107  4.070±  .077  14.52 

OTHER  SERIATIONS  OF  FLORAL  ORGANS:  Ranunculacece. — 
Petals,  Ranunculus  bulbosus,  de  Vries,  '94,  Pearson,  '95; 
calyx,  coralla,  stamens,  and  pistils  of  Ficaria  verna,  Ludwig, 
'01;  number  of  Ficaria  pistils,  early  flowers,  A  =  17.448,  a= 
3.89;  late  flowers,  A  =  12.147,  <r=3.88;  number  of  stamens, 
early,  A  =  26.731,  (7=3.761  and  late,  A  =  17,863,  <y=3.298, 


70  STATISTICAL   METHODS. 

MacLeod,  '99,  Weldon,  '01;  number  of  petals  of  Caltha 
palustris,  de  Vries,  '94;  number  of  calyx  parts  and  petals 
of  Trollius  europseus  and  number  of  fruits  per  head  of  Ranun- 
culus acris,  Ludwig,  '9Sb,  '00b;  number  of  seeds  per  capsule- 
compartment  of  Helleborus  foetidus,  Ludwig,  '97. 

Crucifercc. — Number  of  flowers,  Cardamine  pratensis,  em- 
pirical modes  at  2,  5,  8,  11, 13,  16,  19,  22,  not  in  Fibonacci 
series,  Vogler,  '03. 

Papaveracece. — Number  of  floral  organs  in  Papaver,  Mac- 
Leod, '00;  number  of  sepals  and  petals  in  the  lesser  Celan- 
dine, various  species,  Pearson  and  others,  '03. 

Caryophyllacece. — Number  of  stamens  in  Stellaria  media, 
varies  with  season  and  position  on  plant,  Burkill,  '95;  num- 
ber of  anthers  in  44,542  flowers  of  Stellaria  media — a  com- 
plex polygon  due  to  effect  of  age  and  environment,  Reinohl, 
'03. 

Sapidacece. — Number  of  compartments  in  fruit  of  Acer 
pseudoplatanus,  de  Vries,  '94. 

Leguminosce. — Number  of  blossoms  in  clover  plants,  Type  I: 
(7=2.788,  de  Vries,  '94,  Pearson,  '95,  p.  402;  number  of  ele- 
vated flowers  in  blossoms  of  Trifolium  repens  perumbellatum, 
de  Vries,  '94;  floral  organs  of  Lotus  uliginosus,  L.  cornicu- 
latus,  Medicago  saliva,  M.  falcata,  Ludwig,  '97;  flowers  per 
head  of  Lathyrus,  Ludwig,  '00b. 

Rosacew. — Number  of  stamens  of  Prunus  spino^  a  and  Cra- 
taBgus,  Ludwig,  '01;  sepals  of  1000  Potentilla  lormentilla 
and  petals  of  4097  Potentilla  anserina,  de  Vriep,  '94. 

Cornacece. — Number  of  flowers  in  head  of  Cornus  mas  and 
C.  sanguinea,  not  in  Fibonacci  series,  Vogler,  '03. 

Capri foliacece. — Number  of  petals  of  1167  Weigelea  ama- 
bilis,  de  Vries,  '94:  number  of  flowers  in  inflorescence  and 
number  of  petals  on  flower  of  Adoxa  moschatellina,  White- 
head,  '02. 

DipsaccB. — Number  of  flowers  per  head  in  Knautia  arven- 
sis,  maximum  at  64,  Vogler,  '03. 

Composite. — Number  of  male  and  female  flowers  in  umbel 
of  Homogyne,  Ludwig,  '01. 

Primulacece. — Number  of  flowers  per  umbel,  Primula, 
multimodal,  Ludwig,  '97,  '98b,  '00;  rays  in  Primula  farinosa, 
Vogler,  '01. 


STATISTICAL    BIOLOGICAL    STUDY.  71 

Scrophulariacece. — Number  of  parts  in  peloria  of  Lenaria 
spuria,  Yost,  '99;  number  of  stamens,  Digitalis,  Gallardo,  '00. 

Orchidacece. — Extremes  in  variability  of  number  of  spots 
on  flower,  Chodat,  '01. 

LEAVES. — Seriation  of  numbers  of  paired  leaflets  of  Pirus 
aucuparia,  Fraxinus  excelsior,  Senecio  nemorencis,  and  Pole- 
monium,  Ludwig,  '97,  '98b.  Length  and  breadth  of  leaves 
of  Fagus  silvatica  and  Carpinus  betulus,  Ludwig,  '99.  Leaf- 
dimensions,  Sanguinaria,  Liriodendron,  Ampelopsis,  and 
Ailanthis  (n,  small),  Harshberger,  '01.  Number  of  side  ribs 
on  leaves  of  Fagus  silvatica,  Carpinus  betulus,  and  Quercus 
monticola,  Ludwig,  '99;  on  leaves  of  beech,  Pearson,  '00; 
leaves  of  mulberry,  Fry,  '02;  dimensions  of  Typha  leaves, 
Davenport  and  Blankinship,-'9S;  pine  needles,  Ludwig,  '01; 
from  various  branches  of  Pinus  silvestris,  Lee, -'02. 

Lot>  length  of  pine  needles                  A  mm.  a  mm.                C 

Pinus  silv. ,  lower  branches .  .     22 . 163  db  .  048  4 . 474  ±  .  034  20 . 19 

"     middle  branches .    26. 524  ±.055  5. 167  ±.039  19.48 

"     upper  branches    .   25. 949  ±.062  5. 858  ±.044  22.59 

Fruit. — Number  of  ears  in  head  of  Agropyrum  repens  and 
Brachypodium,  Ludwig,  '01 ;  of  the  grass  Lolium,  Ludwig, 
'00b;  fruits  per  head  of  Ranunculus  acris  Ludwig,  '00b;  num- 
ber of  seeds  per  capsule-compartment,  Helleborus,  Ludwig, 
;97;  fruit  length,  Oenothera  Lamarckiana,  and  Helianthus, 
de  Vries,  '94;  dimensions  of  beans  in  masses  and  in  succes- 
sive generations  of  same  family,  Johannsen,  '03. 

BRYOPHYTA. — Seriations  of  length  of  capsule-stalk,  Bryum 
cirratum,  Amann,  '96;  parts  in  sexual  organs  of  Marchantea 
and  Lonicera,  Ludwig,  '00b. 

SOME  TYPES  OF  BIOLOGICAL  DISTRIBUTIONS. 

General. — Pearson,  '95    'Old.     a  modified  by  selection, 
Reinohl,  '03. 
Type  I. 

Petals  of  222  flowers  of  Ranunculus  bulbosus,  de  Vries,  '94, 
Pearson,  '95,  p.  401. 

Number  of  glands  of  fore  legs  of  swine,  Davenport  and 
Bullard,  '96,  Pearson,  '96,  p.  291:  a=. 311  ±.016. 


72  STATISTICAL   METHODS. 

Fertility  (percentage  of  births  with  one  year  of  marriage) 
of  wives  at  different  ages,  Powys,  '01. 

Rays  in  dorsal  fin  of  Pleuronectes  5 ,  Duncker,  '00. 
"      "  anal     "     "  "  ?,          "          " 

Type  IV. 

Stature    of    St.    Louis   schoolgirls,   Pearson,   '95,   p.   386. 
a=-  0.489. 

Number  of  teeth,  Palsemonetes  varians      Plymouth,  Pear- 
son, '95,  p.  404.     a=  0.134. 

Stature  of  Australian  whites,  Powys,  '01. 
Rays  in  dorsal     fin  of  Pleuronectes,  ? ,  Duncker,  '00. 
"      "  anal         "   "  i(  &          "          ll 

"      "  pectoral  "   "  "  $          "          " 

Type  V. 

Number  of  lips  of  medusa,  P.  pentata,  Mayer,  '01,  Pearson, 
'Old.     a=.549. 

Normal. 

Stature,  U.  S.  recruits,  Baxter,  75,  Pearson,  95,  p.  385. 
Ray  frequency,  Pectens,  Davenport,  '00,  '03b. 

Skewiiess. 

GENERAL. — Mathematical  Analysis. — Pearson,  '95,  'Old,  '02f, 
'02^,  '02ra.     Biological  Interpretation.— Davenport,    ;01b,  'Olc. 

Quantitative  Results. 

Numerous   cranial    characters,    Naquada    race,  Fawcett,   '02. 

Nearly  always  -K 

Num.  lips  of  medusa,  P.  pentata  (Mayer,  '01;  Pearson,  '01(i) +  .549 

Num.  Miillerian  glands,  legs  of  swine  (Pearson  arid  Filon,  '98). .  .  +  .311 

Num.  dorsal  teeth,  Palsemonetes  varians  (Pearson,  '95) +  .  130 

Num.  rays,  Pecten  opercularis,  Irish  Sea  (Davenport,  '031')- .....  +  .087 
Eddystone  (Davenport,  '03h) .  .  .  .  +  .080 

"      hooks  on  statoblasts,  Pectinatella  (Davenport,  '00e) +  .077 

Weldon's  crab  measurements,  "No.  4  "  (Pearson,  '95) +  .077 

Num.  rays  lower  valve,  Pecten  irradians,  L.  I    (Davenport,  '00C)+  .023 

"         "        "  "       P.  opercularis,  F.  of  Forth +  .007 

'    upper  valve,  P.  irradians  (Davenport,  '00C) db  .000 

Height,  British  criminals  (Macdonell,  '02) —  .023 

Baxter's  height  of  U.  S.  recruits  (Pearson,  '95) —  .038 

Porter's  height  of  2192  St.  Louis  schoolgirls  (Pearson,  '95) —  .049 

Head  breadth,  British  criminals  (Macdonell,  '02) -  .051 


STATISTICAL    BIOLOGICAL    STUDY. 

Index  of  Littorina,  Casco  Bay  (Bumpus.  '98) +  .  13 

Index  of  Littorina   Newport  (Bumpus.  '98) -f  .25 

Humber          "  '     +  .048 

So  Kincardineshire  (Bumpus  '98) +  .068 

21-rayed  Chrysanthemum  (de  Vries,  '99) —  .  13 

13-     "  "  "        "         "       +  .12 

Selected  12-  (and  13-)  rayed  Chrysanthemum  (de  Vries,  '99)  ....  +1 .9 

Rays  of  Pectenirradians,  fossil,  Va    oldest  (Davenport,  'Olb)  .  .  .  .  —  .22 

'*     youngest —  .  16 

"      "       "  "          recent,  NC -.09 

•*      "       "  "          recent,  L  I.  . +  .023 

Length  of  wings  of  long-winged  chinch- bug  (Davenport,  'Olb).  ..  —  .43 
"         "       "       "  short-winged  chinch- bug  "     ...+  .44 

Length  horns  rhinoceros-beetle ,  long-horned  (Davenport,  '011}).  .  —  .03 
"  "  "  "        short-horned  "      ..  +  .48 

Complex  Distributions. 

Bimodal  Polygons. — Discontinuity  in  hairiness  of  Biscu- 
tella,  Saunders,  '97;  of  Lychnis,  Bateson  and  Saunders,  '02, 
Weldon,  '02C. 

Length  of  cephalic  horns  of  rhinoceros-beetle,  and  forceps 
length  of  male  earwigs,  Bateson,  '94;  explanation  of  di- 
morphism, Giard,  '94. 

Multimodal  Polygons. — Modes  fall  in  Fibonacci  series,  Lud- 
wig,  '96,  '96b,  '96C,  '97,  '97b,  '97C 

Modes  of  Chrysanthemum  segetum  at  13,  21,  de  Vries, '95. 

Opposed  to  Fibonacci  series,  complex  polygon  due  to  lack 
of  homogeneity,  Lucas,  '93,  Shull,  '02,  Pearson,  '02* ,  Lee,  '02, 
Reinohl,  '03,  Vogler,  '03. 

CORRELATION. 

General  and  Method. — Galton,  '88,  '89,  Pearson, 
'96,  Yule,  '97,  '97b;  spurious  correlation,  Pearson,'97;  non- 
quantitative  characters,  Pearson,  '00C,  Pearson  and  Lee,  '00, 
Yule,  '00,  '00b,  '02;  index  not  constant  in  related  races, 
Weldon,  '92,  Pearson,  '96,  '98b  p.  175,  '02n  p.  2,  Daven- 
port, '03b. 
Man. 

General — Galton,  '88;  British  criminals,  various  dimen- 
sions, r=.13  to  .84,  Macdonell,  '02. 

Skull — Correlated  with  cranial  capacity  in  living  persons, 
Lee  and  Pearson,  '01 ;  breadth  and  length,  Naquada,  Bavari- 
ans, French,  Pearson,  '96,  p.  280;  N.  A.  Indians,  Boas,  '99; 


74  STATISTICAL   METHODS. 

various  dimensions,  Aino  and  German,  Lee  and  Pearson,  '01; 

Naquadas,  Fawcett  and  Lee,  '02.     With  civilization  woman's 

correlation   tends    to  gain  on  man's,  Lee  and  Pearson,  '01, 

Pearson,  '02n. 

Lot.  r 

Breadth  and  Length: 

German,  * 49±  .05 

Smith  Sound  Eskimo 47 

Aino,  a 43  ±  .06 

Aino,  ? 37  ±  .07 

German,  $ 29±.06 

Modern  Bavarian  peasants 28  ±  .  06 

Naquada  race 27 

Sioux  Indians. 24 

Modern  French  peasants 13  ±  .  09 

British  Columbian  Indians  $ 08 

Modern  French  (Parisians) 05±  .06 

Shuswap  Indians 04 

Lot.                                                r$  r? 
Aino: 

Capacity  and  length 89  ±  .  01  . 66  ±  .  05 

11    breadth 56±.05  .50±.07 

"    height 54±.05  .52±.07 

Length  and  height 50±  .05  .35±  .07 

Breadth  and  height 35  ±  .  06  . 18  ±  .  08 

Cap.  and  ceph.  index -.31  ±.07  -.25±.09 

German : 

Capacity  and  breadth 67  d=  .04  .70±  .03 

11    length 51±.05  .69±.04 

"    height 24±.06  .45±.05 

Cap.  and  ceph.  index 20  ±  .  06  -  .  03  ±  .  07 

Breadth  and  height 07  ±  .  06  . 28  db  .  06 

Length  and  height -  .  10±  .07  .31  ±  .06 

Skeletal. — Rollet,  '89;  stature  correlated  with  length  of 
long  bones,  reconstruction  of  stature  of  extinct  races,  Pear- 
son, '98b;  various  coefficients  of  correlation,  Pearson,  '99,  '00, 
p.  402;  in  hand-bones,  Whiteley  and  Pearson,  '99,  Lewenz 
and  Whiteley,  '02. 


STATISTICAL   BIOLOGICAL   STUDY.  75 

Lot.  r 

Right  and  left  femur 95 

Metacarpals,  ii  and  iii  digits  right 94 

First  joints,  iv  digit,  R.  and  L.  hands 93 

First  joints,  ii  and  iii,  right 90 

Metacarpals,  ii  and  v  digits,  right 89 

Femur  and  humerus 84  to  .  87 

Femur  and  tibia 81  to  .  89 

First  joints,  ii  and  v,  right 82 

Stature  and  femur 80(  ? )  to  .  81(  $  ) 

Stature  and  humerus 77(?)  to  .81(4) 

Stature  and  tibia 78 ( a )  to  .  80(  ? ) 

Humerus  and  ulna.  .  . . 75  to  .86 

Humerus  and  radius 74  to  .84 

Radius  and  stature 67  ( « )  to  70(  a ) 

Clavicle  and  humerust 44  to  .  63 

Forearm  and  stature 37 

Clavicle  and  scapula 12  to  .  16 

Stature  and  cephalic  index —  .08 

Various:  Pearson,  '99;  intelligence  not  correlated  with 
size  or  shape  of  head,  Pearson,  '02. 

Weight  and  length  of  new-born  infant  $ 644  ±  .012 

!t     "        "          "        ? 622±.013 

Weight  and  stature  of  Cambridge  (Engl.)  students,  $  .  . .        .486±  .016 

"  "  ? 721  ±.026 

Breadth  of  head  (reduced  to  12th  yr.)  and  intelligence, 

youth 084±  .024 

Length  of  head  (reduced  to  12th  yr.)  and  intelligence, 

youth 044  ±  .  024 

Cephalic  index  and  intelligence,  youth 005  ±  .024 

Breadth  of  head  and  ability,  adults 045  ±  .032 

Cephalic  index  and  ability,  University  men 031  ±  .035 

"    length  of  head,  University  men -  .086±  .033 

Vaccination  and  Recovery. — Pearson,  '00C;  Macdonell,  '02, 
'03.  r=  .23  to  .91. 

Assortative  Mating. — Pearson,  '96,  '99b,  '00,  Pearson  and 
Lee,  '00. 

Stature  of  husbands  and  wives r—  .093±  .047 

ditto,  another  determination r=  .28   ±  .02 

Eye-color,   husbands  and  wives r=  .  100  ±  .038 

Age  at  death  of  consorts r=  .  22 


76  STATISTICAL    METHODS. 

Lower  Animals. 

ANTIMERTCALLY  SYMMETRICAL  ORGANS: 

Paired  organs. — Number  of  Miillerian  glands  on  R.  and  L. 
fore  legs  of  swine,  Davenport  and  Bullard,  '96;  R.  and  L. 
fins  of  fishes,  Duncker,  '97,  '00;  number  of  coxal  pores  on  R. 
and  L.  legs  of  the  centipede  Lithobius,  Williams,  '03;  R.  and 
L.  dimensions  of  Gelasimus,  Yerkes,  '01,  Duncker,  '03;  num- 
ber of  teeth  on  R.  and  L.  jaws  of  Nereis,  Hefferan,  '00; 
breadth  of  R.  and  L.  valves  of  Pecten,  Davenport,  '03b; 
skeletal  spicules  on  R.  and  L.  half  of  Echinus  larva. 

Subject  and  Relative.  r 

Length  R,  and  L.  sides  of  carapace,  Gelasimus 947  ±  .003 

"         "      "     M    meropodite,  first  walking  leg 918  ±  .005 

Breadth  R.  and  L.  valve  of  Pecten  opercularis,  Irish  Sea.  .  .    .858±  .006 

Num.  of  teeth  R.  and  L  jaws  of  Nereis 820  ±  .008 

"       "    fin-rays  R.  and  L  pectoral,  Acerina 710 

"     coxal  pores  R.  and  L.  14th  pair  legs,  Lithobius 69    ±  .  02 

"      "     "    13th  pair  legs,  Lithobius 686±.029 

"      "     "    12th  pair  legs,  Lithobius 58    ±.04 

"          "         "       "     "    "    anal  pair  legs,  Lithobius 575±.039 

Other  antimeric  organs: 

r 
Num    of  dorsal  and  ventral  spines,  Palsemonetes  vulgaris 

(Duncker,  'OQb) 380±  .019 

Num.    of   lips   and   canals   of    the   medusa,    Pseudoclytia 

(Mayer,  '01 ;  Davenport,  '02) 325 ±  .019 

SECONDARILY  ANTIMERIC  ORGANS. — (Median  organs  in 
animals  that  lie  on  one  side.) 

r 
Num,  of  dorsal.and  anal  fin-rays  in  flounder,  $ 651 

"       "       "         "       "          "          "          "          ? 690 

Length  antero-posterior  and  dorso-ventral  diameters,  Pecten  .  970  ±  .001 

Unsymmetrical  paired  organs.  —  Pleuronectes,  Duncker, 
'00;  Gelasimus,  the  fiddler-crab,  Yerkes,  '01,  Duncker,  '03. 

Length  of  meropodite,  R.  and  L.  chelae  of  Galasimus 754  ±  .014 

"        "  carpopodite,  R.  and  L.  chelae  of  Gelasimus 698  ±  .017 

"        "  propodite,  R.  and  L.  chelse  of  Gelasimus 473  i  .026 

Num.  rays  R.  and  L.  pectoral  fin,  flounder,  Pleuronectes,  $  .   .594 
"      "      "     "         "          "  "  "  ?.    .582 

"   of  dorsal  fin-rays  at  which  lateral  line  ends,  R.  and  L. 

Pleuronectes,  $ 467 

Num.  rays  R,  and  L.  ventral  fin,  Pleuronectes,  $ 243 


STATISTICAL   BIOLOGICAL   STUDY.  77 

METAMERICALLY  REPEATED  ORGANS. — Fin-rays  of  fishes, 
Duncker,  '97;  coxal  pores  centipede,  Williams,  '03;  seg- 
ments of  shrimp  Crangon,  Weldon,  '92. 

Num.  dorsal  spines  and  soft  fin-rays,  Acerina —  .379 

"      "        "      Cot  t  us 11C 

"       coxal  pores  R.  anal  and  14th  segment,  Lithobius 440 

R.  13th  and  14th  segments,  Lithobius 722 

R.  13th  and  12th  segments,  Lithobius 464 

Length  carapace  and  post-spinous  portion  rostrum,  Crangon 81 

"    tergum  VI  abd.  seg.,  Crangon 09 

"        tergum  VI  and  telson,  Crangon — .11 

MIXED  AND  CROSS  CORRELATION. — Length  of  wing  and  tail  of 
Lanius  "  shrike,"  Strong,  '01;  in  fishes,  Duncker,  '97,  '99;  pro- 
portions of  aphids,  "plant-lice,"  Warren,  '02;  coxal  pores 
of  centipede,  Williams,  '03;  length  of  carapace  and  of  chela? 
in  Eupagurus,  "hermit-crab,"  Schuster,  '02;  diameter  of 
cell  and  body  length,  Daphnia,  Warren,  ;03;  cross  correla- 
tion in  teeth  on  jaws  of  Nereis,  Hefferan,  '00;  various  char- 
acters of  the  mud-snail,  Nassa,  Dimon,  '02;  circumference  to 
number  of  spines,  statoblast  of  Bryozoa,  Davenport,  '00e; 
diameter  of  body  of  the  Heliozoan  Actinosphaerium  Echorni 
and  the  number  of  cysts  and  of  nuclei,  Smith,  '03;  inner  and 
outer  diameters  and  color  of  the  shell  of  Arcella,  Pearl  and 
Dunbar,  '03. 

Organs .  r 

Carapace  length  and  chela  length,  Eupagurus,  $ 9389  ±   0036 

"  "  ? 8626±.()080 

Diameter  of  body  of  Actinospherian  and  num.  of  nuclei        .854    ±  .  017 

Inner  and  outer  diameter  shell  of  Arcella 836    ±  .007 

Diam.  of  body  of  Actinosphasrium  and  num.  of  cysts.  .         .769   ±  .026 

Wing  length  and  tail  length,  Lanius 569 

Diam.  of  cell  and  body  length,  Daphnia,  hatching  to 

3d  molt 551 

Diam..  of  cell  ai\d  body  length,    Daphnia,  3:1  to  4th 

molt 393 

Diam.  of  cell  and  body  length,  Daphnia,  after  4th  molt..        . 248 

Num.  coxal  pores,  R.  anal  and  L.  12th  seg. ,  Lithobius. .  .         .  427    ±  .  046 

Frontal  breadth  and  antennal  length  (Warren,  '02) 320-  ±  .032 

Ccxal  pores,  R.  14th  leg  and  body  length,  Lithobius..  .        .308   ±  .059 
Num.  rays  dorsal  fin  and  end-point  of  L.  lateral  line, 

Pleuronectes,  $ '.        -208 

Outer  diameter  and  color  Arcella 012 

Num.  dorsal  spines  and  L.  pectoral  rays,  Pleuronectes.        .004 


78  STATISTICAL   METHODS. 

Organs.  r 

Body  length  and  number  antennal  joints —  .013±  .067 

Circumference    of    statoblast     and     number    spines, 

Pectinatella  . —  .092  ±  .006 

Num.  R.  definite  teeth  and  L.  indefinite,  Nereis —  .524  ±  .023 

Carapp-ce  length  and  chela  index,  Eupagurus —  .522±  .022 

Num.  of  cysts  and  their  diam.,  Actinosphserium —  .669±  .040 

Plants. 

Between  various  parts  of  flowers,  Ludwig,  '01. 

Floral  parts. — Stamens  and  pistils  of  Ficaria,  MacLeod, 
'98,  '99,  Ludwig,  '01,  Weldon,  '01,  Lee, '02;  rays  and  bracts 
and  rays  and  disc  florets  of  Aster,  Shull,  '02;  various  organs 
on  Lesser  Celandine,  Pearson  and  others,  '03. 

Organs.  r 

Num.  rays  and  bracts.  Aster 856  to  .799 

stamens  and  pistils  Ficaria  ranunculoides,  early.  ..        .507±    .031 

late 749  ±    .015 

rays  and  disc  florets,  Aster 574  to  .353 

"     petals  and  sepals  Ficaria  verna +  .  34  to  —  .  18 

"     stamens  and  pistils,  Celandine 43  to       .75 

"    petals.  Celandine 38  to       .22 

pistils  and  petals,  Celandine 35  to       .19 

sepals,  Celandine 25  to       .03 

"       stamens  and  sepals,  Celandine 06  to       .02 

Other  parts. — Size  of  leaves  of  same  rosette  of  Bellis  peren- 
nis,  Verschaffelt,  '99;  various  pairs  of  dimensions  of  fruits 
and  leaves,  Harshberger,  '01;  parts  of  Syndesmon,  Keller- 
man,  '01. 

HEREDITY. 
General. 

Treatises.— Galton,  '89,  Pearson,  '00. 

Classification. — Galton,  '89,  pp.  7, 12,  Pearson  and  Lee,  '00, 
pp.  89,  91,  98. 

Law  of  ancestral  heredity. — Galton,  '97,  Pearson,  '98;  esti- 
mate of  heredity  from  a  single  ancestral  generation,  Pearson, 
'96,  p.  306. 

Inequality  in  parental  transmission. — Father  prepotent  in 
sons;  mother  in  daughters,  Pearson  and  Lee,  '00,  p.  115; 
heredity  weakened  by  change  of  sex,  Pearson  and  Lee,  '00, 
p.  115,  Lutz,  '03. 


STATISTICAL   BIOLOGICAL   STUDY. 


79 


Inheritance  of  Eye-color,  Homo, 
s.son;  d,  daughter;  /.father;  w,  mother. 


(  Average  of  rg/  and  rdn 
Parental   •<         ,,         „  rJ    an(jr 


Grand- 
pare: 


(  "  Vandr<* 

intal    1  "t  rsfm'  Tdffi 

'  "  rsm/»  rd/m 


m/^  rsmm  • 
Great-grand-parental  inheritance,  average  .  . 


No .  of  Changes  of  Sex. 


0 
.530 


.370 


.347 


.459 
.300 
.222 


.296 
.145 


.038 


Parental. 

Exceptional  fathers  produce  exceptional  sons  at  a  rate 
three  to  six  times  that  of  non-exceptional  fathers  and  ex- 
ceptional pairs  at  ten  times  the  rate  of  non-exceptional  pairs, 
Pearson,  '00C,  pp.  38,  47. 

x  y  Cor.   Reg. 

Longevity:  r       PXy 

Father  and  son  (Beeton  and  Pearson,  '99) .12 

*'    adult  son  (Beeton  and  Pearson,  '01) .135      .10 

"    adult  dau.      "         "  ' 130     .08 

Mother  and  adult  son     "          "         "  "     .  131      . 12 

••        "      dau.    "         "         "  " 149     .12 

Eye-color  (Pearson  and  Lee,  '00) 55  to  .44 

Stature,  English  middle  class: 

Father  and  son  (Pearson,  '96,  p.  270) 396     .352 

••    dau.         "  "          "        -360      .419 

Mother  anjd  son          "  "          "        302     .269 

"    dau.         "  "          "        284     .275 

Head  index,  N.  Amer.  Indian: 

Mother  and  son  (Pearson, '00,  p.  458) .370 

"    dau.         "  300 

Coat-color,  thoroughbred  horses: 

Sire,  foal  (Pearson.  '00,  p.  458) 

Dam,  foal       "  527 

Fertility : 

Mother  and  daughter,  British  upper  class 

Father  and  son,  -051  ±  -009 

Mother  and  daughter,  British  peerage .210 

Father  and  son,  

Mother  and  daughter,  landed  gentry 

Father  and  son  -116 

r  P 

Frontal  breadth,  Hyalopterus  (Warren,  '02) 

Length  R.  antenna,  Hyalopterus     '  

Ratio:  R.  antenna  +  frontal  breadth  (Warren,  '02)  .  .  . 
Ratio-  Length    protopodite-J- length    body,    Daphnia 

(Warren,  '02) 466     -619 


80 


STATISTICAL   METHODS. 


Graiidpareiital. 

Coat  color,  thoroughbred  race-horses „ 

"       Basset  hounds 

Frontal  breadth,  Hyalopt3rus,  Aohidse  (Warren,  '02) 

Length,  R.  antenna,  Aphidae  (Warren,  '02) 

Ratio     R.  antenna -4- frontai  breadth,  Aphidse  (Warren,  '02 
Ratio    Length   protopodite  -r- length  body,  Daphnia  (War- 
ren. '02) 

Gr'dson  and  gr'df..  homo   male  line  (Pearson,  '96) 

"  "        "  female  l>ne  (Pearson,  '96). 

Ortgr'dson  and  grtgr'df.   homo    $  line 


.339 
.113 
.321 
.177 
.231 


.269 
.192 
.295 


[.27        .5] 


.199 
.089 
.105 
.031 


Eye-color,  homo,  f.,  grandfather,  and  son  (Blanchard,  '03)  .421 

.324 
.380 
.360 
.372 
.359 
.297 
.311 
.272 
.309 
.221 
.204 
.262 
.261 
.318 
.239 


Coat 
Eye 
Coat 
Eye 
Coat 
Eye 
Coat 
Eye 
Coat 
Eye 
Coat 
Eye 
Coat 
Eye 
Coat 


horse, 

homo,  " 

horse,  " 

homo,  m., 

horse,  " 

homo,  " 

horse,  " 

homo,  f.,  grandmother,  and  son 

horse,  " 

homo,  "  "    dau. 

horse,  " 

homo,  m., 

horse,  " 

homo,  " 

horse,  " 


dau. 


son 
dau. 


son 


dau. 


Fraternal.  r 

Daphnia,  length  of  spine  (Warren,  '99;  Pearson,  'Olc) 693 

Aphis,  antennal  length  (Warren,  '02) 679 

"       frontal  breadth  (Warren,  '02) 666 

Paramecium,  index  of  just  separated  fission  pairs  (Simpson,  '02).    .664 
Horse,  coat-color  (Pearson,  Lee,  and  Moore),  average  of  3  sets.  .    .633 

Man,  forearm,  English  (Pearson,  'Olc) 542 

Hound,  coat-color,  Bassett  (Pearson  and  Lee,  '00) 526 

Man,  eye-color,  English  (Pearson,  'Olc).     Average  of  2  sets 475 

Pectinatella,  statoblast  hooks  (Pearson,  '01 c) 430 

Man,  stature  Average  of  3  sets.  .    .403 

cephalic  index,  N.  A.  Ind.  Average  of  3  sets.  .    .403 

"      longevity,  Quakers  (Beeton  and  Pearson,  '01) 332 

*'      temper,  British  (Pearson,  'Olc) 317 

longevity,  British  peerage  (Pearson,  '01) 260 

•*  "  Quakers      '*  "  "     197 

Average  of  23  sets 476 

Mean  of  42  fraternal  correlations  (Pearson,  '02k) 495 

Some    mental    characteristics,    inherited    exactly    like    physical 
characters  (Pearson,  'Ole): 

Conscientiousness  .  . 593     Popularity 504 

Self-consciousness.  . 592     Vivacity 470 

Shyness 528     Intelligence 456 

Average  of  6 507 


STATISTICAL    BIOLOGICAL    STUDY.  81 


Theoretical  coefficient  of  correlation  be- 
tween relatives. — Pearson,  '00,  Pearson  and  Lee,  '00. 

Blended  Alternative 

Inherit-  Inherit- 

ance, ance. 

Offspring  and  Parent 3000  . 5000 

"     grandparent 1500  .250 

"  "     great-grandparent 0750  .123 

"    gt.-gt. -grandparent.  .  .    .0375 
"     nth  order  grandparent   .6X(i)n 

Brothers 4000  .4  to  1 .0 

Half-brothers 2000  .2  to  0.5 

Uncle  and  nephew 1500  .250 

First  cousins 0750 

First  cousins  once  removed 0344 

Second  cousins .  0172 

Third  cousins 0041 

Homotyposis. 

Correlation  in  non-sexual  reproduction,  as  in  production  of 
homologous  uridifferentiated  physiologically  independent 
parts,  Pearson,  'Olc;  criticism,  Bateson,  '01;  reply,  Pearson, 
'021;  rejoinder,  Bateson,  '03;  correlation  between  differen- 
tiated homologous  organs,  Pearson,  '02e. 


Lot. 

Ceteract  ,  Somersetshire  
Hartstongue,  Somersetshire  
Shirley  poppy  Chelsea 

(y    • 

Character.             \ 

.  Lobes  on  fronds  ... 
.  Sori  on  fronds  
Stigmatic  bands 

Var.  to 
rar.  of 
Race. 
.  78 
.  78 
.  79 
79 

Corre- 
lation. 

.631 
.630 
.615 
.611 
.599 
.591 
.570 
.562 
.549 
.533 
.524 
.466 
.416 
.405 
.400 
.396 
.374 
.355 
.273 
.190 
.183 
.173 

.457 

English  onion.  Hampden  

.Veins  in  tunics  .  . 

Holly  Dorsetshire  .  '  
Soanish  chestnut,  mixed  
Beech,  Buckinghamshire  
Papaver  rhooas,  Hampden  

.  Prickles  on  leaves  .  . 
.  Veins  in  leaves  
.Veins  in  leaves  .... 
Stigmatic  bands 

.  80 
.   81 
.  82 
.   83 
84 
.  85 
.   85 
.  89 
,   91 
,   91 
92 
92 
93 
93 
96 
98 
98 
98 

87.4 

Mushroom  Hampden 

Grill  indices 

Papaver  rh<ras,  Quantocks  
Shirley  poppy.  Hampden  
Spanish  chestnut,  Buckinghamshire 
Broom,  Yorkshire  
Ash.  Monmouthshire  
Papaver  rhooas.  Lower  Chilterns.  .  ,. 
Ash,  Dorsetshire     

.Stigmatic  bands.  .  .  . 
.Stigmatic  bands.  .  .  . 
.  Veins  in  leaves  ,  .... 
.Seeds  in  pods  , 
.  Leaflets  on  leaves,  -  .  , 
.Stigmatic  bands  
Leaflets  on  leaves.  -  .  . 

Ash  Buckinghamshire  
Holly.  Somersetshire  
\Vild  ivy,  mixed  localities 

.  Leaflets  on  leaves 
.  Prickles  on  leaves.  .  .  . 
Leaf  indices 

Nigella  hispanica  Slough 

.Seg  of  seed  -capsules. 
.Seg.  of  seed-vessels.... 
.  Members  of  whorls    . 

Malva  rotundi  folia,  Hampden  
Woodruff,  Buckinghamshire  

Mean  of  22  cases  .  .  , 

. 

Bands  of  capsules  of  Shirley  poppies,  msan  of  8  crops  (Pear- 
son, and  others,  '02) 498 

Mean  of  39  cases  of  homotyposis  (Pearson,  '02' ) 499 


82  STATISTICAL   METHODS. 

Mendelism. 

General  Statement.— Mendel,  '66,  de  Vries,  '00,  '00b,  '00C, 
'03,  Correns,  '00,  Davenport,  '01,  Bateson,  '02,  Castle,  '03; 
critical,  Weldon,  '02,  '03,  Pearson,  W. 

Plants.— Correns,  '00,  '00b,  '01,  '02-'02C,  '03-'03C,  de  Vries, 
'02,  '01-'03,  Bateson  and  Saunders,  '02 

Animals. — Echinoids,    Doncaster,    '03;     poultry,    Bateson 
and  Saunders,  '02;  mice,  Darbishire,  '02,  '03,  '03b,  Castle,  '03b, 
Bateson,  '03b;   rabbits,  Woods,  '03. 
Telegoiiy. 

No  evidence  of,  in  human  statures,  Pearson  and  Lee,  '96. 
Fertility. 

Inherited  in  man  and  race-horses,  Pearson,  Lee,  and  Bram- 
ley-Moore,  '99;  greater  fertility  in  poppy  of  seeds  from  cap- 
sules with  a  high  number  of  stigmatic  bands,  Pearson,  '02; 
fertility  of  medusae  with  symmetrical  bands  exceeds  that  of 
the  unsymmetrical  as  3  to  4,  Mayer,  '01. 
SELECTION. 

General. — Intensity  of  selection  connotes  a  lessening  tf 
correlation,  Pearson,  '02d,  p.  23;  mediocre  individuals  not 
the  fittest  to  survive,  Pearson,  '02n,  p.  50. 

Man. — 50%  to  80%  of  human  death-rate  selective,  Beeton 
and  Pearson,  '01. 

Other  Animals. — Annihilation  of  the  extremes  in  the. spar- 
row, Bumpus,  '99;  percentage  death-rate  of  families  of 
Aphids  has  inverse  correlation  with  length  of  antenna  of 
mother  (r= -.201  ±.084),  with  frontal  breadth  of  mother 
(r= —.184  ±.084),  and  with  number  in  newly  born  brood 
(r=  -.188 ±.084);  in  Carcinus  moenas,  Weldon,  '95,  '99; 
in  Clausilia,  Weldon,  '01. 

Plants. — Transformation  of  skew  frequency  curve  to  a  sym- 
metrical one  by  selection,  de  Vries,  '94,  '98;  shifting  of  the 
mode  by  selection,  de  Vries,  '99. 

Sexual. — Pearson,  '96:  A  a 

Stature  of  husbands,  inches 69.136±.126  2.628±.089 

"  males  in  general 69 . 215  ±  .  066  2 . 592  ±  .  047 

"  wives 63.869±  .110  2.303±.078 

"  adult  females  in  general  .  .  64. 043  ±.061  2. 325  ±.043 

See  also  Correlation:    Assortative  mating  (p.  75). 

DISSYMMETRY. 

The  following  values  for  A  have  been  determined  by 
Duncker,  '00  and  '03: 


STATISTICAL   BIOLOGICAL   STUDY.  83 


Pleuronectes  flesus  L.,  1060  R.-eyed  and  60  L.-eyed:        Right-  Left- 
eyed,  eyed. 

Num.  of  pectoral  divided  rays 997  —  .983 

Total  num.  pectoral  rays 604  —  .583 

Num.  of  ventral  divided  rays 326  —  .374 

Total  num.  of  ventral  fin-rays 019  —  .083 

Gelasimus  pugilator  Latr.  (fiddler-cnJt)).                          Right-  Left- 
handed,  handed. 

Lateral  edge  of  carapace 838  .793 

Length  of  meropodite,  first  ambulacral  appendage.    .813  .872 
Length  of  meropodite,  of  carpopodite,  and  of  pro- 

podite  of  chelse,  all 1 .00  1 .00 

Num.  of  rays  on  R.  and  L.  pectoral  fins,  Acer  ina —0.111 

"       "  glands  011  wrists  of  swine .0053 

DIRECT  EFFECT  OF  ENVIRONMENT. 

Animals. — Aphids  reared  in  successive  generations  in  in- 
creasingly unfavorable  conditions  have  reduced  dimensions, 
Warren,  '02: 

Grandmother.    Grandchildren. 

Frontal  breadth,  Aphid.  .    A  =  37 . 56  33 . 93 

Length  of  R.  antenna.  ...   A  =  83 . 91  76 . 59 

Ratio  ^-5-*.  .   A  =  22.46  22.57 

K.  A. 

Depauperization  of  mud-snail,  Nassa,  in  diluted  sea-water, 
Dimon,  ;02. 

Plants.— Conditions  of  life  affect  number  of  floral  parts  in 
poppy,  de  Vrics,  '99,  MacLeod,  '00,  Pearson  and  others,  '03; 
number  of  ray-flowers  of  Primula  farinosa  increases  with 
moisture,  Vogler,  '01 ;  empirical  mode  in  number  of  anthers 
in  Stellaria  in  poor  environment  is  3;  in  good  environment 
5,  Reinohl,  '03;  leaf -blade  smaller  in  light  than  in  shade, 
MacLeod,  '98. 

LOCAL  RACES. 

General. — Davenport  and  Blankenship,  '98,  Davenport,  '99. 
Pisces. — Leuciscus  from  different  altitudes,  Eigenmann, 
'95;  herring  from  different  sea-areas  distinguishable,  Heincke, 
'97,  98;  mackerel  from  three  Scotch  localities  differ,  Wil- 
liamson, '00;  fin-rays  of  Pleuronectes  from  New  England 
shore,  Bumpus,  '98: 

Wood  H  oil.      Waquoit.     Bristol,  R.I. 
Dorsal  fin-i ays.  .  .  A  =  66 . 1          65 . 2          64 . 9 
Anal          "          .  A  =  49.7         48.6          48.7 


84  STATISTICAL    METHODS. 

Number  of  fin-rays  of  Pleuronectes  flesus  from  Western 
Baltic,  Af'=39,  southern  North  Sea  41^,  Plymouth  44, 
Duncker,  '99. 

Fish  in  similar  and  adjacent  lakes  belonging  to  different 
drainage-basins  have  marked  difference  in  scales  on  nape, 
number  of  fin-rays  and  of  dorsal  spines,  Moenkhaus,  '96. 

Invertebrata.  —  Mean  and  variability  of  deep-  and  shallow- 
water  Eupagurus  differ,  Schuster,  '03;  proportions,  variability, 
and   correlation    coefficients   of  Pecten   opercularis  differ  at 
Eddystone,  Irish  Sea,  and  Firth  of  Forth,  Davenport,  '03b. 
Plants.  —  Lesser  celandine,  Pearson  and  others,  '03. 

USEFUL  TABLES. 

Probability  Integral.  —  Area  and  ordinate  of  normal  curve 
in  terms  of  abscissa,  Sheppard,  '93,  '03;  abscissa  of  normal 
curve  in  terms  of  ordinate,  Sheppard,  '93;  abscissa  and  ordi- 
nate in  terms  of  difference  of  area,  Sheppard,  '03;  abscissa 
of  normal  curve  in  terms  of  class  index,  Sheppard,  '98. 

Probability  of  fitted  curve  being  the  true  one: 


__     .  .  .     24fi     ^,_ 

Elderton,  '02. 

Values  of  log  ]  /I/  —  e~~^2  c  for  various  values  of  ^2. 

Elderton,  '02. 

Table  of  log  —.  -  —  -  -  --  .     Elderton,  '02. 

—          — 


Table  of  4/  —  I     e~*x2d%,  for  different  values  of  /,  Elder- 

r      *  J\ 

ton,  '02. 

Table  of  Iog10  (!+#)—  x  Iog10  e  for  various  values  of  a*,  for 
use  with  curves  of  Type  III. 

Tables  for  calculating  probable  error,  Sheppard,  ;98. 

Table  of  values  of  1—  r2  and  \/l—r2  for  all  values  of  r 
from  0  to  1  proceeding  by  hundredths,  Yule,  '97. 

Probable  errors  of  r  for  all  values  of  n,  Yule,  '97. 


BIBLIOGKAPHY.  85 


BIBLIOGRAPHY. 

Note. — An  effort  has  been  made  to  include  all  recent 
works  containing  usable  quantitative  data  in  botany  and 
zoology;  but  the  literature  on  the  mathematical  treatment 
of  statistics  and  that  affording  data  in  anthropology  are 
by  no  means  completely  listed. 

ABBREVIATIONS. 

The  following  names  of  journals  often  referred  to  have 

been  much  abbreviated: 

Amer.  Nat.  =  American  Naturalist. 

Ber.  d.  deutsch.  bot.  Ges.  =  Berichte  der  deutschen  botanischen 
Gesellschaft. 

Biom.=  Biometrika. 

Bot.  Centralbl.  =  Botanisches  Centralblatt. 

Phil.    Trans.  =  Philosophical     Transactions     of     the     Royal 
Society  of  London. 

Proc.  Roy.  Soc.  =  Proceedings  of  the  Royal  Society  of  Lon- 
don. 

The  references  are  scattered  through  fifty-seven  periodi- 
cals. 

ADAMS,  C.  C.  '00.     Variation  in  lo.     Proc.  Amer.  Assoc.  for 

the  Adv.  of  ScL,  XLIX,  18  pp.,  27  plates. 
AGASSIZ,  A.,  and  W.  McM.  WOODWORTH,  '96.     Some  varia- 
tions in  the  Genus  Eucope.      Bull.  Mus.  Comp.  Zool., 

XXX,  123-150.     Plates  I-IX.     Nov. 
ALLEN,  J.  A.,  '71.     On  the  Mammals  and  Winter  Birds  of 

East  Florida,  etc.     Bull.  Mus.  Comp.  Zool.,  II,  161-450. 

Plates  IV-VIII. 
AMANN,  J.,  '96.     Application  du  calcul  des  probability  a 

1' etude   de   la   variation   d'un   type   vegetal.      Bull,    de 

1'Herb.  Bossier.     IV,  578-590. 
AMMON,  OTTO,  '99.     Zur  Anthropologie  der  Badener.     Jena: 

G.  Fischer,  707  pp,  15  Tab. 
BACHMETJEW,  P.,  '03.     Ueber  die  Anzahl  der  Augen  auf  der 

Unterseite    der  Hinterfliigel  von  Epinephele  jurtina  L. 

Allgemeine  Zeitschr.  f.  Entomologie,  VIII,  253-256. 
BAKER,  F.  C.,  '03.     Rib  Variation  in  Cardium.     Amer  Nat., 

XXXVII,  481-488,  July. 
BALLOWITZ,    E.,    '99.     Ueber   Hypomerie    und    Hypermerie 

bei  Aurelia  aurita.     Arch.   f.  Entw.  Mech.   d.   Organis- 

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86  STATISTICAL   METHODS. 

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BATESON,  W.,  '89.  On  some  Variations  of  Cardium  edule 
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BATESON,  W.,  '94.  Materials  for  the  Study  of  Variation. 
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BATESON,  W.,  '01.  Heredity,  Differentiation,  and  Other 
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BATESON,  W.,  and  E.  R.  SAUNDERS,  '02.  Reports  to  the 
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BATESON,  W.,  '03.  Variation  and  Differentiation  in  Parts 
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BAXTER,  J.  H.,  '75.  Statistics,  Medical  and  Anthropological, 
of  the  Provost-Marshal-General's  Bureau.  2  vols.  Wash- 
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BEETON,  MARY,  and  K.  PEARSON,  '99.  Data  for  the  Problem, 
etc.  II.  A  First  Study  of  the  Inheritance  of  Longevity, 
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BEETON,  MARY,  and  KARL  PEARSON,  '01.  On  the  Inheritance 
of  the  Duration  of  Life,  and  on  the  Intensity  of  Natural 
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BEETON,  M.,  G.  U.  YULE,  and  K.  PEARSON,  '00.  Data  for  the 
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BIGELOW,  R.  P.,  and  ELEANOR  P.  RATHBUN,  '03.  On  the 
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BURKTLL,  I.  H.,  '95.  On  some  Variations  in  the  Number  of 
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BYRNE,  L.  W.,  '02.  On  the  Number  and  Arrangement  of 
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EXPLANATION    OF   TABLES.  105 


EXPLANATION  OF  TABLES. 

I.  Formulas.    In  this  table  the  principal  formulas  used 
in  the  calculation  of  curves  are  brought  together  for  con- 
venient reference.     The  meanings  of  the  letters  are  explained 
in  the  text.     This  table  is  preceded  by  an  index  to  the  prin- 
cipal letters  used  in  the  formulae  of  this  book. 

II.  Certain  constants   and    their   logarithms. 

This  table  includes  the  constants  most  frequently  employed 
in  the  calculations  of  this  book. 

III.  Table  of  orclinates  of  normal  curve.    Th's 
table  is  for  comparison  of  a  normal  frequency  polygon  con- 
sisting of  weighted  ordiriates  with  the  theoretical  curve. 

Example:  4  =  17.673;        a=  1.117;        y0=181.4. 
(See  page  26.) 

Entries  in  Table 
V  —  M    corresponding  to 


V 

14 
15 
16 

V  —M 

-3.673 
-2.673 
-1.673 

a 

3.29 
2.39 
1.50 

V  —M 

2/0 

X181.4  = 
X181.4  = 
X181.4  = 

y 

0.8 
10.4 
58.9 

; 

1 
8 
63 

a 

.00449 
.05750 
.32465 

IV.    Table   of  values   of  probability  integral. 

This  table  is  for  comparison  of  a  normal  frequency  polygon 
consisting  of  rectangles  with  the  theoretical  curve. 
Example:  A  =  17.673;  <r  =  1.1169.     (See  page  26.) 


Deviation 

(£_£a)Xll 

Class. 

X 

Per 

Class 

from 

Xi 

r»oo 

a 

cent. 

Limits. 

A=XI 

a 

less  2,  x  _._ 

14 

-3.29 

.2 

.225 

14.5 

-3.173 

-2.841 

15 

-2.39 

1.6 

2.364 

15.5 

-2.173 

-1.945 

16 

-1.50 

12.4 

12.097 

16.5 

-1.173 

-1.050 

17 

-    .60 

30.3 

29.155 

17.5 

-0.173 

-0.155 

18 

.29 

32.3 

33.194 

18.5 

0.827 

0.740 

19 

1.19 

18.9 

17.873 

19.5 

1.827 

1.636 

20 

2.08 

3.9 

4.524 

20.5 

2.827 

2.531 

21 

2.98 

0.4 

.568 

100.0 

100.000 

106  STATISTICAL   METHOPS. 

In  the  example,  the  data  of  which  are  given  on  p.  26,  the 

frequency  between  the  limits  is  given  in  %  column.    The  —  of 

a 

each  limit  (as  an  inner  class  limit)  is  found  and  the  entries 
in  Table  IV  corresponding  to  the  limits  are  taken.  Each 
such  entry  is  subtracted  from  0,50000,  'is  multiplied  by 
100,  and  from  the  product  is  subtracted  the  total  theoretical 
percentage  of  variates  lying  between  the  outer  limit  of  the 
class  and  the  corresponding  extremity  of  the  half  curve. 
This  gives  the  theoretical  frequency  of  the  class  in  question. 
The  closeness  of  agreement  of  the  last  column  with  the 
"  Per  cent."  column  indicates  the  closeness  of  the  observed 
frequency  to  the  theoretical. 

V.  Table  of  log  T  functions  of  p.  This  table 
will  enable  one  to  solve  the  equations  for  yQ  given  on  page  32. 
The  table  gives  the  logarithms  of  the  values  of  F  functions 
only  within  the  range  p  =  1  to  2.  As  all  values  of  the  func- 
tion within  these  limits  are  less  than  1,  the  mantissa  of  the 
logarithms  is  —  1;  but  it  is  given  in  the  table  as  10  —  1  =  9, 
as  is  usually  done  in  logarithmic  tables. 

Supposing  the  quantity  of  which  we  wish  to  find  the  value 
reduced  to  the  form  JT(4.273).  The  value  cannot  be  found 
directly  because  the  value  of  p  is  larger  than  the  numbers  in 
the  table  (1  to  2).  The  solution  is  made  by  aid  of  the  equation 


log  T(l.  273)  =  9.  955185 
log      1.273  =0.104828 

log  1X2.273)  =  0.060013 
log      2.273  =0.356599 

log  T(3.273)  =  0.416612 
log     3.273  =0.514946 


log  T(4.273)  =  0.931558 

or,  more  briefly,  log  T(1.273)  =  9.955185 
log  1.273  =  .104828 
log  2.273  =  .356599 
log  3.273  =  .514946 

log  T(4.273)  =  0.931558  =  log  8.542 


EXPLANATION    OF   TABLES.  107 

VI.  Table  of  reduction  from  the  common  to 
the  metric  system.    This  is  given  first  for  whole  inches 
from  1  to  99  excepting  even  tens,  which  may  be  got  from  the 
first  line  of  figures  by  shifting  the  decimal  point  one  place 
to  the  right.     The  table  may  be  used  for  hundredths  of  an 
inch  by  shifting   the  decimal  point  two  places  to  the  left. 
Other  fractious  than  decimals  are  given  in  the  lower  tables. 

VII.  Table  of  minutes  and  seconds  of  arc  in 
decimals  of  a  degree.     This  table  will  be  found  of  use 
in  the  fitting  of  curves  of  Type  IV  (p.  33). 

VIII.  First  to  sixth  powers  of  integers  from  1 
to  3O.     This  table  is  useful  in  calculating  moments. 

IX.  Table  of  the  probable  errors  of  the  coeffi- 
cient of  correlation  for  various  numbers  of  ob- 
servations or  variates  (ri)  and  for  various  values 
of  r.     The  probable  error  of  the  coefficient  of  correlation 

,    .      0.6745(1 -r2)  f 

being  —      — := -,  a  table  for  the  varying  values  of  n  and  r 

Vn 

is  easily  constructed,  and  for  large  values  of  n  is  accurate 
with  interpolation  by  inspection  to  two  significant  figures, 
which  are  all  that  are  required. 

X.  Squares,  cubes,  square  roots,   and  recip- 
rocals  of  numbers  from    1  to  1O54.    The  use  of 
this  table  can  be  extended  by  using  the  principle  that  if  any 
number  be  multiplied  by  n,  its  square  is  multiplied  by  n2,  its 

cube  by  n3,  and  its  reciprocal  by  — . 

XI.  Logarithms    of   numbers    to    six    places. 

The  following  explanation  of  the  use  of  the  logarithmic  tables 
-is  taken  from  Searles'*  Field  Engineering,  pp.  257-263   [ed. 
1887]. 

The  logarithm  of  a  number  consists  of  two  parts, 
a  whole  number,  called  the  characteristic,  and  a  decimal, 
called  the  mantissa.  All  numbers  which  consist  of  the 
same  figures  standing  in  the  same  order  have  the  same  man- 
tiss"a,  regardless  of  the  position  of  the  decimal  point  in  the 
number,  or  of  the  number  of  ciphers  which  precede  or  follow 
the  significant  figures  of  the  number.  The  value  of  the  char- 
&cteristic  depends  entirely  on  the  position  of  the  decimal  point 


108  STATISTICAL   METHODS. 

iii  the  number.  It  is  always  one  less  than  the  number  of 
figures  in  the  number  to  the  left  of  the  decimal  point.  The 
value  is  therefore  diminished  by  one  every  time  the  decimal 
point  of  the  number  is  removed  one  place  to  the  left,  and  vice 
versa.  Thus 

Number.  Logarithm. 

13840.  4.141136 

1384.0  3.141136 

138.40  2.141136 

13.84  1.141138 

1.384  0.141136 

.1384  —1.141136 

.01384  —2.141136 

.001384  —3.141136 

etc.  etc. 

The  mantissa  is  always  positive  even  when  the  characteristic 
is  negative.  We  may  avoid  the  use  of  a  negative  characteristic 
by  arbitrarily  adding  10,  which  may  be  neglected  at  the  close 
of  the  calculation.  By  this  rule  we  have 

Number.  Logarithm. 

1.384  0.141136 

.1384  9.141136 

.01384  8.141136 

.001384  7.141136 

etc.  etc. 

No  confusion  need  arise  from  this  method  in  finding  a  number 
from  its  logarithm;  for  although  the  logarithm  6.141136  repre- 
sents either  the  number  1,384,000,  or  the  decimal  .0001384,  yet 
these  are  so  diverse  in  their  values  that  we  can  never  be  uncer- 
tain in  a  given  problem  which  to  adopt. 

TABLE  XI,  contains  the  mantissas  of  logarithms,  car- 
ried to  six  places  of  decimals,  for  numbers  between  1  and  9999, 
inclusive.  The  first  three  figures  of  a  number  are  given  in  the 
first  column,  the  fourth  at  the  top  of  the  other  columns.  The 
first  two  figures  of  the  mantissa  are  given  only  in  the  second 
column,  but  these  are  understood  to  apply  to  the  remaining 
four  figures  in  either  column  following,  which  are  comprised 
between  the  same  horizontal  lines  with  the  two. 

If  a  number  (after  cutting  off  the  ciphers  at  either  end)  con- 
sists of  not  more  than  four  figures,  the  mantissa  may  be  taken 
direct  from  the  table ;  but  by  interpolation  the  logarithm  of  a 
number  having  six  figures  may  be  obtained.  The  last  column 
contains  the  average  difference  of  consecutive  logarithms  on 


EXPLANATION   OF   TABLES.  109 

the  same  line,  but  for  a  given  case  the  difference  needs  to  be 
verified  by  actual  subtraction,  at  least  so  far  as  the  last  figure 
is  concerned.  The  lower  part  of  the  page  contains  a  complete 
list  of  differences,  with  their  multiples  divided  by  10. 

To  find  the  logarithm  of  a  number  having  six 
figures  :— Take  out  the  mantissa  for  the  four  superior  places 
directly  from  the  table,  and  find  the  difference  between  this 
mantissa  and  the  next  greater  in  the  table.  Add  to  the  man- 
tissa taken  out  the  quantity  found  in  the  table  of  proportional 
parts,  opposite  the  difference,  and  in  the  column  headed  by  the 
fifth  figure  of  the  number;  also  add  T*o  the  quantity  in  the  col- 
umn headed  by  the  sixth  figure.  The  sum  is  the  mantissa 
required,  to  which  must  be  prefixed  a  decimal  point  and  the 
proper  characteristic. 

Example.—  Find  the  log  of  23.4275. 

For  2342  mantissa  is  369587 

"    diff.  185  col.  7  129.5 

"       "       "     "    5  9.2 

Ans.  For  23.4275  log  is    L369726 

The  decimals  of  the  corrections  are  added  together  to  deter- 
mine the  nearest  value  of  the  sixth  figure  of  the  mantissa. 

To  find  the  number  corresponding  to  a  given 
logarithm. — If  the  given  mantissa  is  not  in  the  table  find  the 
one  next  less,  and  take  out  the  four  figures  corresponding  to  it; 
divide  the  difference  between  the  two  mantissas  by  the  tabu- 
lar difference  in  that  part  of  the  table,  and  annex  the  figures  of 
the  quotient  to  the  four  figures  already  taken  out.  Finally, 
place  the  decimal  point  according  to  the  rule  for  characteristics, 
prefixing  or  annexing  ciphers  if  necessary.  The  division  re- 
quired  is  facilitated  by  the  table  of  proportional  parts,  which 
furnishes  by  inspection  the  figures  of  the  quotient. 

Example. — Find  the  number  of  which  the  logarithm  is 
8.263927  8.263927 

First  4  figures  1836  from  263873 

Diff.          54J) 
Tabular  diff.  =236          ,  • .  5th  fig.  =2  47. 2 


6.80 

6th  fig.  =  3  7.08 

Ans.  No.  =  .0183623  or  183,623,000. 


110  STATISTICAL   METHODS. 

The  number  derived  from  a  six-place  logarithm  is  not 
reliable  beyond  the  sixth  figure. 

At  the  end  of  Table  XI  is  a  small  table  of  logarithms  of 
numbers  from  1  to  100,  with  the  characteristic  prefixed,  for 
easy  reference  when  the  given  number  does  not  -exceed  two 
digits.  But  the  same  mantissas  may  be  found  in  the  larger 
table. 

TABLE  XII. — The  logarithmic  sine,  tangent,  etc., 
of  an  arc  is  the  logarithm  of  the  natural  sine,  tangent,  etc.,  of 
the  same  arc,  but  with  10  added  to  the  characteristic  to  avoid 
negatives.  This  table  gives  log  sines,  tangents,  cosines,  and 
cotangents  for  every  minute  of  the  quadrant.  With  the 
number  of  degrees  at  the  left  side  of  the  page  are  to  be  read 
the  minutes  in  the  left-hand  column;  with  the  degrees  on 
the  right-hand  side  are  to  be  read  the  minutes  in  the  right-hand 
column.  When  the  degrees  appear  at  the  top  of  the  page  the 
top  headings  must  be  observed,  when  at  the  bottom  those  at 
the  bottom.  Since  the  values  found  for  arcs  in  the  first  quad- 
rant are  duplicated  in  the  second,  the  degrees  are  given  from 
0°  to  180°.  The  differences  in  the  logarithms  due  to  a  change 
of  one  second  in  the  arc  are  given  in  adjoining  columns. 

To  find  the  log. sin,  cos,  tan,  or  cot  of  a  given 
arc. :  Take  out  from  the  proper  column  of  the  table  the  log- 
arithm corresponding  to  the  given  number  of  degrees  and 
minutes.  If  there  be  any  seconds  multiply  them  by  the  ad- 
joining tabular  difference,  and  apply  their  product  as  a  cor- 
rection to  the  logarithm  already  taken  out.  The  correction  is 
to  be  added  if  the  logarithms  of  the  table  are  increasing  with 
the  angle,  or  subtracted  if  they  are  decreasing  as  the  angle  in- 
creases. In  the  first  quadrant  the  log  sines  and  tangents  in- 
crease, and  the  log.  cosines  and  cotangents  decrease  as  the 
angle  increases. 

Exampk.— Find  the  log  sin  of  9°  28'  20". 

Log  sin  of  9°  28'  is  9.216097 

Add  correction  20  X  12.62  252 

Ans.  9.216349 
Example.—  Find  the  log  cot  of  9°  28'  20". 

Log  cotan  of  9°  28'  is  10.777948 

Subtract  correction  20  X  12.97  259 

Ans.  ~10777689 


To  find  the  angle  or  arc  corresponding  to  a 
given  logarithmic  sine,  tangent,  cosine,  or  co- 
tangent.—If  the  given  logarithm  is  found  in  the  proper 
column  take  out  the  degrees  and  minutes  directly;  if  not,  find 
the  two  consecutive  logarithms  between  which  the  given 
logarithm  would  fall,  and  adopt  that  one  which  corresponds  to 
the  least  number  of  minutes;  which  minutes  take  out  with  the 
degrees,  and  divide  the  difference  between  this  logarithm  and 
the  given  one  by  the  adjoining  tabular  difference  for  a  quo- 
tient, which  will  be  the  required  number  of  seconds. 

With  logarithms  to  six  places  of  decimals  the  quotient  is 
not  reliable  beyond  the  tenth  of  a  second. 

Example.—  9.383731  is  the  log  tan  of  what  angle? 
Next  less  9.383682  gives  13°  36' 

Diff.  49.00  -*-  9.20  ~  05'.3 


Ans.     13°  36'  05".3 

Example.—  9.249348  is  the  log  cos  of  what  angle? 
Next  greater  583  gives  79°  46' 

Diff,  235  -J- 11.67  =  20M 


Ans.     79°  46  20M 

The  above  rules  do  not  apply  to  the  first  two  pages  of  this 
table  (except  for  the  column  headed  cosine  at  top)  because 
here  the  differences  vary  so  rapidly  that  interpolation  made  by 
them  in  the  usual  way  will  not  give  exact  results. 

On  tbe  first  two  pages,  the  first  column  contains  the  number 
of  seconds  for  every  minute  from  1'  to  2° ;  the  minutes  are 
given  in  the  second,  the  log.  sin.  in  the  tJiird,  and  in  the  fourth 
are  the  last  three  figures  of  a  logarithm  which  is  the  difference 
between  the  log  sin  and  the  logarithm  of  the  number  of  sec- 
onds in  the  first  column.  The  first  three  figures  and  the  char- 
acteristic of  this  logarithm  are  placed,  once  for  all,  at  the  head 
of  the  column. 

To  find  the  log  sin  of  an  arc  less  than  2°  given 
to  seconds. — Reduce  the  given  arc  to  seconds,  and  take  the 
logarithm  of  the  number  of  seconds  from  the  table  of  loga- 
rithms, and  add  to  this  the  logarithm  from  the  fourth  column 
opposite  the  same  number  of  seconds.  The  sum  is  the  log  sin. 
required. 


STATISTICAL   METHODS. 

The  logarithm  in  the  fourth  column  may  need  a  slight  inter- 
polation of  the  last  figure,  to  make  it  correspond  closely  to  the 
given  number  of  seconds. 

Example.— Find  the  log  sin  of  1°  39'  14".  4. 

1°  39'  14".4  =  5954".4  log  3.774838 

add  (q  - 1)  4.685515 

Ans.  log  sin  8.460353 

Log  tangents  of  small  arcs  are  found  in  the  same  way,  only 
taking  the  last  four  figures  of  (q  —  I)  from  the  fifth  column. 

Example.—  Find  the  log  tan  of  0°  52'  35". 

52'  35"  =  (3120"  4-  35")  =  3155"  log  5.498999 

add  (q  -  I)  4.685609 

Ans.     log  tan  8. 184608 

To  find  the  log  cotangent  of  an  angle  less  than 
2°  given  to  seconds.— Take  from  the  column  headed  ( q-{-  Z) 
the  logarithm  corresponding  to  the  given  angle,  interpolating 
for  the  last  figure  if  necessary,  and  from  this  subtract  the  loga- 
rithm of  the  number  of  seconds  in  the  given  angle. 

Example.— Find  the  log  cotan  of  1°  44'  22". 5. 

q  +  I  15.314292 
6240"  +  22".  5  =  6262.5  log    3.796748 

Ans.     11.517544 

These  two  pages  may  be  used  in  the  same  way  when  the 
given  angle  lies  between  88°  and  92°,  or  between  178°  and  180°; 
but  if  the  number  of  degrees  be  found  at  the  bottom  of  the  page, 
the  title  of  each  column  will  be  found  there  also;  and  if  the 
number  of  degrees  be  found  on  the  right  hand  side  of  the  page, 
the  number  of  minutes  must  be  found  In  the  right  hand  col- 
umn, and  since  here  the  minutes  increase  upward,  the  number 
of  seconds  on  the  same  line  in  the  first  column  must  be  dimin- 
ished by  the  odd  seconds  in  the  given  angle  to  obtain  the  num- 
ber whose  logarithm  Is  to  be  used  with  (q±l)  taken  from  the 
table. 

Example.—  Find  the  log  cos  of  88°  41'  12". 5 

fe  -  J)  4.685537 
4740"  -  12".5  =  4727.5  log  3.674631 

Ans.  8.360168 


EXPLANATION   OF   TABLES.  113 

Example. — Find  the  log  tan  of  90°  30'  50". 

q  +  I  15.314413 
1800"  +  50"  =  1850*  log    3.267172 

ATM.     12.047241 

To  find  the  arc  corresponding-  to  a  given  log 
sin,  cos,  tan,  or  cotan  which  falls  within  the 
limits  of  the  first  two  pages  of  Table  X. 

Find  in  the  proper  column  two  consecutive  logarithms  be- 
tween which  the  given  logarithm  falls.  If  the  title  of  the 
given  function  is  found  at  the  top  of  that  column  read  the 
degrees  from  the  top  of  the  page;  if  at  the  "bottom  read  from 
the  bottom. 

Find  the  value  of  (q  —  1}  or  (q  -f-  0,  as  the  case  may  require, 
corresponding  to  the  given  log  (interpolating  for  the  last  figure 
if  necessary).  Then  if  q  =  given  log  and  I  —  log  of  number  of 
seconds,  n,  in  the  required  arc,  we  have  at  once  I  =  q  —  (q  —  1} 
or  I  —  (q  -[-  0  —  q,  whence  n  is  easily  found. 

Find  in  the  first  column  two  consecutive  quantities  between 
which  the  number  n  falls,  and  if  the  degrees  are  read  from 
the  left  hand  side  of  the  page,  adopt  the  less,  take  out  the 
minutes  from  the  second  column,  and  take  for  the  seconds 
the  difference  between  the  quantity  adopted  and  the  number 
n.  But  if  the  degrees  are  read  from  the  right  hand  side  of  the 
page,  adopt  the  greater  quantity,  take  out  the  minutes  on  the 
same  line  from  the  right-hand  column,  and  for  the  seconds 
take  the  difference  between  the  number  adopted  and  the  num- 
ber n. 

Example.— 11.734268  is  the  log  cot  of  what  arc? 
q  +  I  15.314376 

q  11.734668 

.-.     n  =  3802.8  "3.580108 

For  1°  adopt      3780.        giving  03' 

Difference  22".  8 

Ans.  1°  03'  22".8  or  178°  56'  37".2. 

J&cample. — 8.201795  is  the  log  cos  of  what  arc? 
q  -  I  4.685556 

q  8.201795 

/.       n  =t  3282".  8  3.51623d 

For  89°  adopt      3300.       giving  05r 

Difference  17  ".2 

Ans,  89°  05'  17".2  or  90°  54'  42". 8. 


114 


STATISTICAL   METHODS. 


THE  GREEK  ALPHABET. 


A  a 

Alpha 

/  i 

Iota 

Pp 

Rho 

B/3 

Beta 

K  K 

Kappa 

3  cr< 

>  Sigma 

r  y 

Gamma 

A  A 

Lamba 

T  r 

Tau 

A  d 

Delta 

M/t 

Mu 

TV 

Upsilon 

E    € 

Epsilon 

N  v 

Nu 

<P  (f) 

Phi 

z  c 

Zeta 

S  1 

Xi 

X  x 

Chi 

Hr; 

Eta 

Oo 

Omicron 

W  0 

Psi 

GQ$ 

••  Theta 

nit 

Pi 

£L  GO 

Omega 

EXPLANATION    OF   TABLES. 


115 


INDEX  TO  THE  PRINCIPAL  LETTERS  USED  IN  THE 
FORMULAE   OF   THIS  BOOK. 


A,  average,  mean. 

a,  class  index  (p.  24);    also  upper 

left-hand  quadrant  (p.  49). 
a,  skewness  index. 

6,  the     frequency   of   the    upper 
right  quadrant  (p.  49). 

/?,  ratio  of  moments. 

C,  coefficient  of  variability. 

c,  the  frequency  of  the  lower  left 
quadrant  (p.  49). 

D,  distance  from  mean  to  mode. 

d,  a  difference;    differential;    the 
frequency  of  lower  right  quad- 
rant (p.  49). 

4,  index  of  closeness  of  fit. 

d,  difference  between  y  and  /. 

E,  probable  error. 

e,  base    of    Naperian   logarithms, 
1      =2.718282. 

F,  critical  function. 
/,  class  frequency. 

G,  geometric  mean. 
H,  a  function  of  h. 

h,  a  fixed  value  of  x\  also,  index  of 
heredity. 

7,  interval  between  the  p'th  and 
j>"th  individual. 

t,  interval  between  the  pth   and 

(p  +  l)th  individual  (p.  27). 
K,  a  function  of  k. 
k,  a  fixed  value  of  x. 
L,  limiting  value  of  class. 
Z,  range  of  curve  along  x. 
l\,  ^2,  portions  of  the  curve  range. 
A ,  number  of  classes. 
A,  class  range. 
M ,   abscissal   value   of  the   mode 

(theoretical). 
M',  abscissal  value  of  the  mode 

(empirical). 
H,  moment  about  A. 
N,  the  number  corresponding  to 

a  log. 


n,  number   of   variates;    area  of 

polygon;     any,     not     specified, 

number. 
|rc_,  product    of    all    integers  from 

1  to  n. 

v,  average  moment  about  VQ. 
H,  index  of  dissymmetry. 
P,  probability 
p,   ordinal    rank    of    a    particular 

individual  or  case    (p.    27);    a 

root  or  power. 
ic,  circumference  in  units  of  diame- 

ter, 3.14159. 
q,  a  root  or  power. 
r,  coefficient  of  correlation. 
p,  coefficient  of  regression. 
s,  a  relation  of  @'s  (p.  22). 
2",  summation  sign. 
<?,    standard   deviation;    index   of 

variability. 
T,  transmuting  factor,   o  into  E, 

.67449. 
r,  in  Type  IV. 

'  f  angles. 

9»     ) 

V,  magnitude  of  any  class. 

FO,  magnitude  of  central  class. 

v,  any  variate  or  value. 

™  =  5/?2-6/?i-9  (p.  31). 

X,  the  horizontal  axis  or  base  of 

polygon. 

x,  a  varying  abscissal  value. 
x\,  X2,  etc.,  definite  values  of  x. 


Y,  the  vertical  axis  of  polygons  ; 

also  the  log  of  /  (p.  29). 
y,  a  varying  ordinate  value. 
2/0,  value  of  the  ordinate  at  the 

origin. 
z,  ordinate  value. 


116  STATISTICAL   METHODS. 

I.  FORMULAS. 


=  ±0.6745—^=.     x  =  V-J. 
\/n 

=  0.6745-^, 

V2n 


C=j-XlOO%.  E, 

A.  D.=^^  =  0.7979(7.  ^       =0.6745(7. 


1  I 


(*4./)  ,   j  ^(rr2./)  ,     7    j. 
71^^2/1     ^240  J  * 


£|f  (Types  I,  IV).  a  =  2Vp     3  (Type  y)> 

Probable  discrepancy,  ™^'  \  ±  . (-^} -  A+^    I  * 
Vn      /   2         2/  \        J7    ) 

J(dev.  rcXdev.  yX/)  =  S(x&2f)  _  0.6745(1  -r2) 

(72 
ro  (spurious  correlation)  = — — — 


. 

71          o2  n 

To  solve  any  equation  of  the  second  degree, 

-b±\/b2-4ac 
Q;     x= =~ . 


CERTAIN   CONSTANTS   AND   THEIR   LOGARITHMS.      117 
II.— CERTAIN  CONSTANTS  AND  THEIR  LOGARITHMS. 


Title. 

Symbol. 

7T 
1 

7T 

VT 

1 

VT 

\/2ir 
1 

\/2~x 
1 

27T 

VT 

1 

V2~ 

V? 

e 
1 
VT 
m 

1 
m 

T 

Number. 

Log. 

Ratio  of  circumference  to  diameter 

3.1415927 
0.3183099 

1.7724538 
0.5641896 
2.506628 
0.3S89422 

0.159155 
1.4142136 
0.707105 
0.797816 

2.7182818 
0.606530 
0.4342945 
2.3025851 
0.67449 

0.4971499 
9.502CSC1 

0.248574i 
9.75K-2fl 
0  .  3C909G 
9.6009101 
9.201820 
0.150515 
9.8494849 
9.9019401 

0.4342945 
9.78285281 
9.6377843 
0.3622157 
9.828976 

Reciprocal  of  same  .          .            

Reciprocal  of  sQiiare  root  of  same  . 

Reciprocal  of  same  ...       .      ... 

Reciprocal  of  2?r  .  .        

Square  root  of  2  

Reciprocal  of  same       .             ... 

2 
Square  root  of  —  

7T 

Base  of  hyperbolic  logarithms  

Reciprocal  of  square  root  of  same 

Modulus  of  common  system  of  logs  =  log  e 
Reciprocal  of  same  —  hyp.  log  10 

Factor  to  reduce  a  to  probable  error.  .  .  . 

Com.  log:r  =  raXhyp.  log  x,  or 

Com.  log  (com.  log  x) 
=  9.6377843  +  com.  log  (hyp.  log*) 

Hyp.  log  x  =  com.  logo:  X—  ,  or 
m 

Com.  log(hyp.  logs) 
=  com.  log  (com.  log)  x  +  0.  3622157 

Circumference  of  circle  

2xr 
«r« 

lAlr 
a       9 
360^ 

where  a 
semi-n 

=  semi-majoi 
linor  axis  of 

•  axis:    6  = 
ellipse. 

Area  of  circle  

Area  of  sector  (length  of  arc  =Z)  

Area  of  sector  (angle  of  arc  =a°)  

F            '  *t       f           I!'               A/a2  —  &2 

a2 

118 


STATISTICAL   METHODS. 


TABLE  III.— TABLE  OF  ORDINATES  (2)   OF  NORMAL  CURVE, 

OR  VALUES  OF  —  CORRESPONDING  TO  VALUES  OF  — .     • 

yb  a 

y=  frequency. 
2/o  =  — 7=  = maximum  frequency. 


x  =  deviation  from  mean. 
a  =  standard  deviation. 


X/a 

0 

1 

2 

3 

4 

99920 
99025 
97161 
94387 
90774 

5 

99875 
98881 
96923 
94055 
90371 

6 

99820 
98728 
96676 
93723 
89961 

7 

8 

9 

0.0 
0.1 
0.2 
0.3 
0.4 

100000 
99501 
98020 
95600 
92312 

99995 
99396 
97819 
95309 
91939 

99980 
99283 
97609 
95010 
91558 

99955 
99158 
97390 
94702 
91169 

99755 
98565 
96420 
93382 
89543 

99685 
98393 
96156 
93034 
89119 

99596 
98211 
95882 
92677 

88688 

0.5 
0.6 
0.7 
0.8 
0.9 

88250 
83527 
78270 
72615 
66698 

87805 
83023 
77721 
72033 
66097 

87353 
82514 
77167 
71448 
65494 

86896 
82010 
76610 
70861 
64891 

86432 
81481 
76048 
70272 
64287 

85962 
80957 
75484 
69681 
63683 

85488  85006 
80429  79896 
74916  74342 
69087  68493 
63077  62472 

84519 
79359 
73769 
67896 
61865 

84060 
78817 
73193 
67298 
61259 

.0 
.1 
.2 
.3 

.4 

60653 
54607 
48675 
42956 
37531 

60047 
54007 
48092 
42399 
37007 

59440 
53409 
47511 
41845 
36487 

58834 
52812 
46933 
41294 
35971 

58228 
52214 
46357 
40747 
35459 

57623 
51620 
45783 
40202 
34950 

57017 
51027 
45212 
39661 
34445 

56414 
50437 
44644 
39123 
33944 

55810 
49848 
44078 
38589 
33447 

55209 
49260 
43516 
38058 
32954 

.5 
.6 

.7 
.8 
.9 

32465 
27804 
23575 
19790 
16448 

31980 
27361 
23176 
19436 
16137 

31500 
26923 
22782 
19086 
15831 

31023 
26489 
22392 
18741 
15530 

30550 
26059 
22008 
18400 
15232 

30082 
25634 
21627 
18064 
14939 

29618 
25213 
21251 
17732 
14650 

29158 
24797 
20879 
17404 
14364 

28702 
24385 
20511 
17081 
14083 

28251 
23978 
20148 
16762 
13806 

2.0 
2.1 
2.2 
2.3 
2.4 

13534 
11025 

08892 
07100 
05614 

13265 
10795 
08698 
06939 
05481 

13000 
10570 
08507 
06780 
05350 

12740 
10347 
08320 
06624 
05222 

12483 
10129 
08136 
06471 
05096 

12230 
09914 
07956 
06321 
04973 

11981 
09702 
07778 
06174 
04852 

11737 
09495 
07604 
06029 
04734 

11496 
09290 
07433 

05888 
04618 

11259 
09090 
07265 
05750 
04505 

2.5 
2.6 
2.7 
2.8 
2.9 

04394 
03405 
02612 
01984 
01492 

04285 
03317 
02542 
01929 
01449 

04179 
03232 
02474 
01876 
01408 

04074 
03148 
02408 
01823 
01367 

03972 
03066 
02343 
01772 
01328 

03873 
02986 
02280 
01723 
01289 

03775 
02908 
02218 
01674 
01252 

03680 
02831 
02157 
01627 
01215 

03586 
02757 
02098 
01581 
01179 

03494 
02684 
02040 
01536 
01145 

3 
4 
5 

01111 
00034 
00000 

00819 
00022 

00598 
00015 

00432 
00010 

00309 
00006 

00219 
00004 

00153 
00003 

00106 
00002 

00073 
00001 

00050 
00001 

VALUES    OF    NORMAL   PROBABILITY    INTEGRAL.      119 


TABLE  IV.-TABLE  OF  THE  HALF  CLASS  INDEX  (*a)  VALUES 
OF  THE  NORMAL  PROBABILITY  INTEGRAL  CORRESPOND- 
ING TO  VALUES  OF  -;  OR  THE  FRACTION  OF  THE  AREA 

a 

OF  THE   CURVE   BETWEEN  THE   LIMITS  0  AND   +-,  OR  0 

a 

AND  -— . 

a 

Total  area  of  curve  assumed  to  be  100,000. 
x  =  deviation  from  mean. 
a  —  standard  deviation. 


x/a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

J 

0.00 

00000 

40 

80 

120 

159 

199 

239 

279 

319 

359 

40 

0.01 

0399 

439 

479 

519 

559 

598 

638 

678 

718 

758 

0.02 

0798 

838 

878 

917 

957 

997 

1037 

1077 

1117 

1157 

0.03 

1197 

1237 

1276 

1316 

1356 

1396 

1436 

1476 

1516 

1555 

0.04 

1595 

1635 

1675 

1715 

1755 

1795 

1834 

1874 

1914 

1954 

0.05 

1994 

2034 

2074 

2113 

2153 

2193 

2233 

2273 

2313 

2352 

0.06 

2392 

2432 

2472 

2512 

2551 

2591 

2631 

2671 

2711 

2751 

0.07 

2790 

2830 

2870 

2910 

2949 

2989 

3029 

3069 

3109 

3148 

0.08 

3188 

3228 

3268 

3307 

3347 

3387 

3427 

3466 

3506 

3546 

0.09 

3586 

3625 

3665 

3705 

3744 

3784 

3824 

3864 

3903 

3943 

0.10 

3983 

4022 

4062 

4102 

4141 

4181 

4221 

4261 

4300 

4340 

0.11 

4380 

4419 

4459 

4498 

4538 

4578 

4617 

4657 

4697 

4736 

0.12 

4776 

4815 

4855 

4895 

4934 

4974 

5013 

5053 

5093 

5132 

0.13 

5172 

5211 

5251 

5290 

5330 

5369 

5409 

5448 

5488 

5527 

0.14 

5567 

5606 

5646 

5685 

5725 

5764 

5804 

5843 

5883 

5922 

0.15 

5962 

6001 

6041 

6080 

6119 

6159 

6198 

6238 

6277 

6317 

0.16 

6356 

6395 

6435 

6474 

6513 

6553 

6592 

6631 

6671 

6710 

0.17 

6750 

6789 

6828 

6867 

6907 

6946 

6985 

7025 

7064 

7103 

0.18 

7142 

7182 

7221 

7260 

7299 

7338 

7378 

7417 

7456 

7495 

0.19 

7535 

7574 

7613 

7652 

7691 

7730 

7769 

7809 

7848 

7887 

0.20 

7928 

7965 

8004 

8043 

8082 

8121 

8160 

8199 

8238 

8278 

0.21 

8317 

8356 

8395 

8434 

8473 

8512 

8551 

8590 

8628 

8667  39 

0.22 

8706 

8745 

8784 

8823 

8862 

8901 

8940 

8979 

9018 

9057 

0.23 

9095 

9134 

9173 

9212 

9250 

9289 

9328 

9367 

9406 

9445 

0.24 

9483 

9522 

9561 

9600 

9638 

9677 

9716 

9754 

9793 

9832, 

0.25 

9871 

9909 

9948 

9986 

10025 

10064 

10102 

10141 

10180 

10218 

0.26 

10257 

10295 

10334 

10372 

10411 

10449 

10488 

10526  10565 

10603' 

0.27 

10642 

10680 

10719 

10757 

10796 

10834 

10872 

10911  10949 

10988 

0.28 

11026 

11064 

11103 

11141 

11179 

11217 

11256 

11294 

11333 

11371 

0.29 

11409 

11447 

11485 

11524 

11562 

11600 

11638 

11676 

11715 

11753 

0.30 

11791 

11829 

11867 

11905 

11943 

11981 

12019 

12058 

12096 

12134 

0.31 

12172 

12210 

12248 

12286 

12324 

12362 

12400 

12438 

12476 

12514 

38 

0.32 
0.33 

12552 
12930 

12589  12627 
12968  13005 

12665 
13043 

12703 
13081 

12741 
13118 

12778 
13156 

12816 
13194 

12854 
13232 

12892 
13269 

0.34 

13307 

13344  13382 

13420 

13457 

13495 

13533 

13570 

13608 

13645 

0.35 

13683 

13720 

13758 

13795 

13833 

13870 

13908 

13945 

13983 

14020 

PROPORTIONAL  PARTS. 

A 

1 

2345 

6 

789 

40 

4.0 

8.0   12.0   16.0   20.0 

24.0 

28  .  0   32  .  0   36  .  0 

39 

3.9 

7.8   11.7   15.6   19.5 

23.4 

27.3   31.2   35.1 

38 

3.8 

7.6   11.4   15.2   19.0 

22.8 

26.6   30.4   34.2 

37 

3.7 

7.4   11.1   14.8   18.5 

22.2 

25.9   29.6   33.3 

120 


STATISTICAL   METHODS. 


TABLE  IV.— Continue* 


x/a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

A 

0.36 

14058 

14095 

14132  14169 

14207 

14244 

14281 

14319 

14356 

14393 

0.37 

14431 

14468 

14, 

305  145< 

12 

14579 

14617 

14654 

14691 

14 

728 

147 

65 

0.38 

14803 

14840 

14* 

*77  149 

14 

14951 

14988 

15025 

15062 

15 

099 

151 

36 

37 

0^39 

15173 

15210 

15247 

15284 

15321 

15357 

15394 

15431 

15468 

15505 

0.40 

15542 

15579 

15( 

516 

156 

v_> 

15689 

15726 

15763 

15799 

15 

836 

158 

73 

0.41 

15910 

15946 

15983 

16019 

16056 

16093 

16129 

16166 

16202 

16239 

0  42 

16276 

16312 

16: 

548 

163! 

35 

16421 

16458 

16494 

16531 

16 

567 

166 

04 

0.43 

16640 

16676 

16713 

16749 

16785 

16821 

16858 

16894 

16930 

16967 

0  44 

17003 

17039 

17( 

)75 

171 

11 

17147 

17184 

17220 

17256 

17 

292 

173 

28 

0.45 

17364 

17400 

17436 

17472 

17508 

17544 

17580 

17616 

17652 

17688 

36 

0.46 

17724 

17760 

17' 

r96 

178 

31 

17867 

17903 

17939 

17975 

IS 

Oil 

18C 

46 

0.47 

18082 

18118 

18153 

18189 

18225 

18260 

18296 

18332 

18367 

18403 

0.48 

18439 

18474 

18, 

509 

185^ 

15 

18580 

18616 

18651 

18687 

18 

722 

187 

58 

0  49 

18793 

18829 

18* 

364 

188< 

39 

18934 

18969 

19005 

19040 

IS 

075 

191 

11 

0.50 

19146 

19181 

1C216 

19251 

19287 

19322 

19357 

19392 

19427 

19462 

0.51 
0.52 

19497 
19847 

19532 
19881 

19567 
19916 

19602  19637 
19951  119986 

19672 
20020 

19707 
20055 

19742 
20090 

19777 
20125 

19812 
20160 

35 

0.53 

20194 

20229 

20263 

20298 

20332 

20367 

20402 

20436 

20471 

20505 

0.54 

2054020574 

20( 

509 

206 

13 

20678 

2C712 

20746 

20781 

2C 

815 

20* 

$50 

0.55 

20884  20918 

20952 

20986 

21021 

21055 

21089 

21123 

21158 

21192 

0.56 

2122621260 

2L 

294 

213 

28 

21362 

21396 

21430 

21464 

21 

498 

211 

532 

34 

0.57 

21566  21600 

21634 

21667 

21701 

21735 

21769 

21803 

21836 

21870 

0.58 

21904  21938 

21< 

371 

220 

35 

22039 

22072  22106 

22139 

^ 

>173 

22S 

>()7 

0.59 
0.60 
0.61 
0.62 

22240122274 
225751  22608 
22907  j  22940 
23237  23270 

22307  22341 
22641  22674 
22973  23006 
23303  '23335 

22374 
22707 
23039 
23368 

22407  22441 
22741  22774 
23072  23105 
23401  23434 

22474 
22807 
23138 
23467 

22508 
22840 
23171 
23499 

22541 
22874 
23204 
23532 

33 

0.63 

23565  23598 

23630  23663 

23695 

23728  23761 

23793 

23826 

23859 

0.64 

23891  23924 

23 

356 

239 

S8 

24021 

24053  24085 

24118 

2^ 

U50 

24 

L83 

0.65 

24215  24247 

24 

280 

243 

12 

24344 

24376  24408 

24441 

2^ 

t473 

24, 

SOS 

0.66 

24537 

24569 

24601 

24633 

24665 

24697  24729 

24761 

24793 

24825 

32 

0.67 

24857 

24889 

24 

920 

249 

->2 

24984 

25016  25048 

25079 

2, 

5111 

25 

143 

0.68 

25175 

25206 

25238  25269 

25301 

25332  25364 

25395 

25427 

25459 

0.69 

25490 

25521 

25 

,;>:; 

255 

84 

25615 

25647 

25678 

25709 

2, 

5741 

25' 

111 

0.70 

25804 

25835 

25 

S66 

258 

97 

25928 

25959 

25990 

26021 

2( 

3052 

26( 

384 

0.71 

26115 

26146 

26176 

26207 

26238 

26269 

26300 

26331 

26362 

26393 

31 

0.72 

26424 

26454 

26 

485 

265 

16 

26546 

26577 

26608 

26638 

2( 

1661 

26' 

roo 

0.73 

26730 

26761 

26791 

26822 

26852 

26883 

26913 

26943 

26974 

27004 

0.74 

27035 

27065 

27 

095 

271 

25 

27156 

27186 

27216 

27246 

2' 

^277 

27 

307 

0.75 

27337 

27367 

27 

397 

274 

27 

27457 

27437 

27517 

27547 

2' 

r577 

27( 

307 

30 

0.76 

27637 

27667  :  27697 

27726 

27756 

27786 

27816 

27845 

27875 

27905 

0.77 

27935 

27964 

27 

994 

280 

23 

28053 

28082 

28112 

28142 

2i 

3171 

28 

201 

0.78 

28230 

28260 

28289 

28318 

28347 

28377 

28406 

28435 

28465 

28494 

0.79 

28524 

28553 

28 

582 

286 

11 

2864C 

28669 

28698 

28727 

2* 

3756 

28 

rss 

0.80 

28814 

28843 

28872 

28901 

2893C 

28958 

28987 

29016 

29045 

29074 

29 

PROPORTIONAL,  PARTS. 

J 

1 

2 

3 

4 

5 

6 

7 

8 

9 

37 

3.7 

7.4 

11.1 

14.8 

18.5 

22.2 

25.9 

29.6 

33.3 

36 

3.6 

7.2 

10.8 

14.4' 

18.0 

21.6 

25.2 

28.8 

32.4 

35 

3.5 

7.0 

10.5 

14.0 

17.5 

21.0 

24.5 

28.0 

31.5 

34 

3.4 

6.8 

10.2 

13.6 

17.0 

20.4 

23.8 

27.2 

30.6 

33 

3.3 

6.6 

9.9 

13.2 

16.5 

19.8 

23.1 

26.4 

29.7 

32 

3.2 

6.4 

9.6 

12.8 

16.0 

19.2 

22.4 

25.6 

28.8 

31 

3.1 

6.2 

9.3 

12.4 

15.5 

18.6 

21.7 

24.8 

27.9 

30 

3.0 

6.0 

9.0 

12.0 

15.0 

18.0 

21.0 

24.0 

27.0 

29 

2.9 

5.8 

8.7 

11.6 

14.5 

17.4 

20.3 

23.2 

26.1 

YALUES   OF   NORMAL   PROBABILITY    INTEGRAL.      121 


TABLE  IV.— Continued. 


x/a 

Q 

1 

2 

3 

4 

5 

6 

7 

8 

9 

J 

0.81 
0.82 

29103  \ 
29389  \ 

29132 
29417 

29160 
29446 

29189 
29474 

29217 
29502 

29246 
29531! 

29274  \ 
29559M 

29303 

29588 

29332 

29616 

29360 
29645 

0.83 

29673  \ 

29701 

297 

29 

>7 

29785 

29814J29842 

29870 

29898 

299 

.>() 

0.84 

29954  \ 

29982 

30010 

30038 

30066 

30094 

30122 

30150 

30178 

30206 

28 

0.85 

30234  30261 

302 

89 

3031 

30344 

30372 

30400 

30427 

30455 

304 

83 

0.86 

30510  , 

30538 

30£ 

)(>5 

305? 

)3 

J0620 

30648 

30675 

30702 

30730 

307 

57 

0.87 

30785 

30812 

30839 

30866 

30894 

30921 

30948 

30975 

31002 

31030 

0.88 

31057 

31084 

311 

11 

311C 

,s 

31165 

31192 

31219 

31246 

31273 

313 

)() 

27 

0.89 

31327 

31353 

31380 

31407 

31433 

31460 

31487 

31514 

31540 

31567 

0.90 

31594 

31620 

316 

>47 

316' 

ra 

31700 

31726 

31753 

31780 

31806 

318 

;_' 

0.91 

31859 

31885 

31911 

31937 

31964 

31990 

32016 

32042 

32069 

32095 

0:92 

32121 

32147 

321 

73 

321' 

)9 

32225 

32251 

32277 

32303 

32329 

323 

5.5 

26 

0.93 

32381 

32407 

32^ 

L33 

324, 

59 

32484 

32510 

32536 

32562 

32587 

326 

13 

0.94 

32639 

32665 

32690 

32715 

32741 

32766 

32792 

32818 

32843 

32869 

0.95 

32894  32919 

32< 

)45 

329' 

T() 

32995 

33021 

33046 

33071 

33096 

331 

22 

0.96 

33147:33172 

331 

L97 

332S 

l-l 

33247 

33272 

33297 

33322 

33347 

333 

73 

25 

0.97 

33398  33422 

33447 

33472 

33497 

33521  33546 

33571 

33596  33621 

0.98 

33646  '33670 

33( 

595 

337 

19 

33744  33768  33793 

33817 

33842  338 

(17 

0.99 

33891 

33915  33940 

33964 

33988  34013 

34037 

34061 

34086 

34110 

1.00 

34134 

34158(34 

L82 

342( 

)0 

34230 

34255 

34279 

34303 

34327 

343 

51 

24 

1.01 

34375 

34399  34423 

34446 

34470 

34494 

34518 

34542 

34566 

34590 

1.02 

34613 

34637 

34( 

li.i 

346? 

$4 

34708 

34731 

34755 

34778 

34802 

348 

26 

1.03 

34849 

34873 

34896 

34919 

34943 

34966 

34989 

35013 

35036 

35059 

1.04 

35083 

35106 

35 

129 

351 

>2 

35175 

35198 

35221 

35245 

35268 

352 

91 

23 

1.05 

35314 

35337 

35360 

35382 

35405 

35428 

35451 

35474 

35497 

35520 

1.06 

35543 

35565 

35 

588 

356 

10 

35633 

35656 

35678 

35701 

35724 

357 

40 

1.07 

Irjo 

35769 

OCTQQO 

35791 

35814 

35836 

35858 

35881 

35903 

35926 

35948 

35970 

.  Uo 

ooyyo 

01  ^ 

037 

059 

081 

103 

125 

1J.8 

170 

192 

22 

Y.09 

38214 

U1O 

236 

258 

280 

302 

324 

345 

367 

389 

411 

1.10 

433 

455 

477 

4 

is 

520 

541 

563 

585 

607 

6 

>2S 

1.11 

650 

671 

693 

714 

735 

757 

778 

OQ1 

800 

821 

843 

1  .  12 

864 

885 

tJUU 

928 

94i. 

970 

yyi 

012 

034 

055 

1.13 

37176   097 

118 

139 

160 

181 

202 

223 

244 

265 

21 

1.14 

286   306 

327 

348 

368 

389 

410 

430 

451 

472 

1.15 

493   513 

534 

5 

-.1 

574 

595 

615 

636 

656,  ( 

>77 

1.16 

697 

718 

738 

758 

778 

798 

819 

839 

859 

880 

1  .  17 

900 

92C 

94( 

96( 

yol 

ooc 

02C 

04C 

06C 

080 

2C 

1.18 

38100 

120 

139 

159 

17S 

199 

218 

238 

258 

278 

1.19 

298 

317 

337 

3 

r,( 

37e 

395 

415 

434 

454 

t 

i": 

1.20 

493 

512 

531 

551 

57C 

589 

609 

628 

647 

667 

PROPORTIONAL  PARTS. 

A 

1 

2 

3 

4 

5 

6 

7     8 

9 

29 

2.9 

5.8 

8.7 

11.6 

14.5 

17.4 

20.3   23.2 

26.1 

28 

2.8 

5.6 

8.4 

11.2 

14.0 

16.8 

19.6   22.4 

25.2 

27 

2.7 

5.4 

8.1 

10.8 

13.5 

16.2 

18.9   21.6 

24.3 

26 

2.6 

5.2 

7.8 

10.4 

13.0 

15.6 

18.2   20-8 

23.4 

25 

2.5 

5.0 

7.5 

10.0 

12.5 

15.0 

17.5   20.0 

22.5 

24 

2.4 

4.8  • 

7.2 

9.6 

12.0 

14.4 

16.8   19.2 

21.6 

23 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.1   18.4 

20.7 

22 

2.2 

4.4 

6.6 

8.8 

11.0 

13.2 

15.4   17.6 

19.8 

21 

2.1 

4.2 

6.3 

8.4 

10.5 

12.6 

14.7   16.8 

18.9 

20 

2.0 

4.0 

6.0 

8.0 

10.0 

12.0 

14.0   16.0 

18.0 

19 

1.9 

3.8 

5.7 

7.6 

9.5 

11.4 

13.3   15.2 

17.1 

122 


STATISTICAL   METHODS. 


TABLE  IV. — Continued. 


x/a 

0 

1 

2 

3 

4 

5 

6 

800 
990 

7 
819 

8 

9 

J 

1.21 
1.22 

1.23 

1.24 
1.25 
1.26 
1.27 
1.28 

1.29 
1.30 
1.31 
1.32 
1.33 
1.34 

.35 
.36 
.37 
.38 
.39 
.40 

.41 
.42 
.43 
.44 
.45 
.46 
.47 

.48 
.49 
.50 
.51 
.52 
.53 
.54 
.55 

.56 
.57 

.58 
.59 

38686 
876 

705 
895 

724 
914 

743 
933 

762 
952 

139 
324 
507 

688 
866 

042 
216 
388 
557 
725 
889 

052 
213 
371 
527 
683 
834 
984 

781 
971 

158 
343 
525 

706 
884 

060 
233 
405 
574 
742 
906 

838 

857 

19 

18 
17 

16 
15 

14 

13 
12 

008 
195 
380 
562 
742 
920 

095 
268 
439 
608 
775 
938 

027 
214 
398 
580 
760 
937 

112 
285 
456 
625 
792 
955 

117 
277 
434 
590 
744 
894 

046 
232 
417 
598 
778 
955 

39065 
251 
435 
617 
796 
973 

084 
270 
453 
634 
813 
990 

102 
288 
471 
652 
831 

121 

306 
489 
670 
849 

177 
361 
544 
724 
902 

077 
251 
422 
591 
758 
922 

008 
182 
354 
524 
692 
857 

025 
199 
371 
540 
709 
873 

130 
303 
473 
641 
808 
971 

40147 
320 
490 
658 
825  j 
987 

41149 
308 
466 
621 
774 
924 

42073 
220 
364 
507 
647 
785 
922 

165 
337 
507 
675 
841 

004 
165 
324 
481 
637 
789 
939 

088 
234 
378 
521 
661 
799 
935 

020 
181 
340 
497 
652 
804 
954 

102 
248 
393 
535 
675 
813 
949 

083 
215 
345 
473 
599 
724 
846 
967 

036 
197 
355 
512 
667 
819 
969 

117 
263 
407 
549 

688 
826 
962 

068 
229 
387 
543 
698 
849 
998 

084 
245 
403 
558 
713 
864 

101 
261 
418 
574 

728 
879 

133 
292 
450 
605 
759 
909 

013 
161 
306 
449 
591 
730 
867 

002 
136 
267 
396 
524 
649 
773 
894 

028 
175 
321 
464 

605 
744 
881 

016 
149 
.280 
409 
536 
662 
785 
906 

043 
190 
335 

478 
619 
758 
895 

029 
162 
293 
422 
549 
674 
797 
919 

038 
156 
271 
385 
498 

058 
205 
350 
492 
633 
772 
908 

043 
175 
306 
435 
562 
687 
810 
931 

050 
167 
283 
397 
509 

131 
277 
421 
563 
702 
840 
975 

109 
241 
371 
498 
624 
748 
870 
990 

109 
225 
340 
453 

146 
292 
435 
577 
•  716 
854 
989 

122 
254 
383 
511 
637 
760 
882 

43056 
189 
319 
448 
574 
699 
822 
943 

069 
202 
332 
460 
587 
711 
834 
955 

096 
228 
358 
486 
612 
736 
858 
978 

002 
120 
237 
351 
464 

014 
132 
248 
363 
475 

026 
144 
260 
374 
486 

44062 
179 
295 
408 

074 
191 
306 
419 

085 
202 
317 
430 

097 
214 
329 
.  442 

PROPORTIONAL  PARTS. 

J 

1 

2     3 

4 

5 

6 

7     8     9 

19 
18 
17 
16 
15 
14 
13 
12 
11 

1.9 
1.8 
1.7 
1.6 
1.5 
1.4 
1.3 
1.2 
1.1 

3.8    5.7 
3.6    5.4 
3.4    5.1 
3.2    4.8 
3.0    4.5 
2.8    4.2 
2.6    3.9 
2.4    3.6 
2.2    3.3 

7.6 
7.2 
6.8 
6.4 
6.0 
5.6 
5.2 
4.8 
4.4 

9.5 
9.0 
8.5 
8.0 
7.5 
7.0 
6.5 
6.0 
5.5 

11.4 
10.8 
10.2 
9.6 
9.0 
8.4 
7.8 
7.2 
6.6 

13.3   15.2   17.1 
12.6   14.4   16.2 
11.9   13.6   15.3 
11.2   12.8   14.4 
10.5   12.0   13.5 
9.8   11.2   12.6 
9.1   10.4   11.7 
8.4    9.6   10.8 
7.7    8.8    9.9 

VALUES   OF   KORMAL   PROBABILITY   INTEGRAL.      123 


TABLE  IV —Continued. 


x/a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

J 

1.60 

44520 

531 

542 

553 

564 

575 

586 

597 

608 

619 

11 

1.61 

630 

641 

652 

662 

673 

684 

695 

706 

717 

727 

1.62 

738 

749 

760 

770 

781 

791 

802 

813 

823 

834 

1.63 

845 

855 

866 

876 

887 

897 

908 

918 

929 

939 

1  .64 

950 

960 

97  C 

980 

991 

001 

on 

022 

032 

042 

1.65 

45053 

063 

073 

083 

093 

103 

114 

124 

144 

1.66 

154 

164 

174 

184 

194 

204 

214 

224 

234 

244 

10 

1.67 

254 

264 

274 

283 

293 

303 

313 

323 

332 

342 

1.68 

352 

362 

371 

381 

391 

400 

410 

419 

429 

439 

1.69 

449 

458 

467 

477 

486 

496 

505 

515 

524 

534 

1.70 

543 

553 

562 

571 

581 

590 

599 

609 

618 

627 

1.71 

637 

646 

655 

664 

673 

682 

692 

701 

710 

719 

1.72 

728 

737 

746 

755 

764 

773 

782 

791 

800 

809 

9 

1.73 

818 

827 

836 

845 

854 

863 

871 

880 

889 

898 

1.74 

907 

916 

924 

933 

942 

950 

959 

968 

977 

985 

1.75 

994 

003 

on 

020 

028 

037 

045 

054 

062 

071 

1.76 

46080 

088 

096 

105 

113 

121 

130 

138 

147 

155 

1.77 

164 

172 

180 

188 

196 

205 

213 

221 

230 

238 

1.78 

246 

254 

262 

270 

279 

287 

295 

303 

311 

319 

1.79 

327 

335 

343 

351 

359 

367 

375 

383 

391 

399 

8 

1.80 

407 

415 

423 

430 

438 

446 

454 

462 

%  469 

477 

1.81 

485 

493 

500 

508 

516 

523 

531 

539 

547 

554 

1.82 

562 

570 

577 

585 

592 

600 

607 

615 

622 

630 

1.83 

638 

645 

652 

660 

667 

674 

682 

689 

697 

704 

1.84 

712 

719 

726 

733 

741 

748 

755 

762 

770 

777 

1.85 

784 

791 

798 

806 

813 

820 

827 

834 

841 

849 

1.86 

856 

863 

870 

877 

884 

891 

898 

905 

912 

919 

7 

1.87 

926 

933 

939 

946 

953 

960 

967 

974 

981 

988 

1  .88 

QQK 

yyo 

nm 

nns 

nl  ^ 

n9i 

O9S 

035 

040 

fl4Q 

055 

1.89 

47062 

UUl 

069 

UUo 

075 

uio 

082 

LWl 

088 

U^o 

095 

102 

\jt& 
108 

U^*7 

115 

122 

1.90 

128 

135 

141 

148 

154 

161 

167 

174 

180 

187 

1.91 

193 

200 

206 

212 

219 

225 

231 

238 

244 

251 

1.92 

257 

263 

270 

276 

282 

288 

294 

301 

307 

313 

1.93 

320 

326 

332 

338 

344 

350 

356 

362 

369 

375 

1.94 

381 

387 

393 

399 

405 

411 

417 

423 

429 

435 

6 

1.95 

441 

447 

453 

459 

465 

471 

476 

482 

488 

494 

1.96 

500 

506 

512 

517 

523 

529 

535 

541 

546 

552 

1,97 

558 

564 

569 

575 

581 

586 

592 

598 

603 

609 

1.98 

615 

620 

626 

631 

637 

643 

648 

654 

659 

665 

1.99 

670 

676 

681 

687 

692 

698 

703 

709 

714 

719 

2.00 

725 

730 

735 

741 

746 

752 

757 

762 

768 

772 

2.01 

778 

784 

789 

794 

799 

804 

810 

815 

820 

826 

2.02 

831 

836 

841 

846 

851 

856 

862 

867 

872 

877 

2.03 

882 

887 

892 

897 

902 

907 

912 

917 

922 

927 

5 

2.04 

932 

937 

942 

947 

952 

957 

962 

967 

972 

977 

PROPORTIONAL  PARTS. 

A 

1 

2345 

6 

7 

8     9 

11 

1.1 

2.2    3.3   4.4   5.5 

6.6 

7.7 

8.8    9.9 

10 

1.0 

2.0    3.0    4.0    5.0 

6.0 

7.0 

8.0   9.0 

9 

0.9 

1.8    2.7    3.6   4.5 

5.4 

6.3 

7.2   8.1 

8 

0.8 

1.6    2.4    3.2    4.0 

4.8 

5.6 

6.4   7.2 

7 

0.7 

1.4    2.1    2.8    3.5 

4.2 

4.9 

$.6   6.3 

6 

0.6 

1.2    1.8    2.4    3.0 

3.6 

4.2 

4.8    5.4 

124 


STATISTICAL   METHODS. 


TABLE  IV.— Continued. 


X/a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

a 

2rvt 

47QR9 

007 

QQ1 

QQft 

.  UO 

1  (Ma£ 

ys  / 

yy  l 

yyo 

noi 

onfi 

m  i 

mr 

H9n 

(\f)K 

2.06 

48030 

035 

039 

044 

UU-l- 

049 

uuo 
054 

ui  i 

058 

o 

063 

u^u 
068 

\JAO 

073 

2.07 

077 

082 

087 

091 

096 

100 

105 

110 

114 

119 

2.08 

124 

128 

133 

137 

142 

146 

151 

155 

160 

165 

2.09 

169 

173 

178 

182 

187 

191 

196 

200 

205 

209 

2.10 

g!4 

218 

222 

227 

231 

235 

240 

244 

248 

253 

2.11 

257 

261 

266 

270 

274 

27.8 

283 

287 

291 

295 

2.12 

300 

304 

308 

312 

316 

320 

325 

329 

333 

337 

2.13 

341 

345 

350 

354 

358 

362 

366 

370 

374 

378 

2.14 

382 

386 

390 

394 

398 

402 

406 

410 

414 

418 

4 

2.15 

422 

426 

430 

434 

438 

442 

446 

450 

453 

457 

2.16 

461 

465 

469 

473 

477 

480 

484 

488 

492 

496 

2.17 

500 

503 

507 

511 

515 

518 

522 

526 

530 

533 

2.18 

537 

541 

544 

548 

552 

555 

559 

563 

566 

570 

2.19 

574 

577 

581 

584 

588 

592 

595 

599 

602 

606 

2.20 

610 

613 

617 

620 

624 

627 

631 

634 

638 

641 

2.21 

645 

648 

652 

655 

658 

662 

665 

669 

672 

676 

2.22 

679 

682 

686 

689 

692 

696 

699 

702 

7C6 

709 

2.23 

713 

716 

719 

722 

726 

'  729 

732 

736 

739 

742 

2.24 

745 

749 

752 

755 

758 

761 

765 

768 

771 

774 

2.25 

778 

781 

784 

787 

790 

793 

796 

799 

803 

806 

2.26 

809 

812 

815 

818 

821 

824 

827 

83C 

833 

837 

2.27 

840 

843 

846 

849 

852 

855 

858 

861 

864 

867 

3 

2.28 

870 

872 

875 

878 

881 

884 

887 

890 

893 

896 

2.29 

899 

902 

905 

907 

•  910 

913 

916 

919 

922 

925 

2.30 

928 

930 

933 

936 

939 

942 

944 

917 

950 

953 

2.31 

956 

958 

961 

964 

966 

969 

972 

975 

977 

980 

2qo 

983 

OCA 

988 

QQ1 

QQ4 

QQA 

999 

.  •  >  — 

yoO 

yy  i 

yy^ 

yyo 

nr>9 

_ 

2.33 

49010 

012 

015 

017 

020 

023 

025 

\J(jz 

028 

004 
031 

007 
033 

2.34 

036 

038 

041 

043 

046 

048 

051 

054 

056 

059 

2.35 

061 

064 

066 

069 

071 

074 

076 

079 

081 

084 

2.36 

086 

089 

092 

094 

096 

098 

101   103 

106 

108 

2.37 

111 

113 

115 

118 

120 

122 

125   127 

130 

132 

2.38 

134 

137 

139 

141 

144 

146 

148 

151 

153 

155 

2.39 

158 

160 

162 

164 

167 

169 

171 

173 

176 

178 

2.40 

180 

182 

185 

187 

189 

191 

193 

196 

198 

200 

2.41 

202 

205 

207 

209 

211 

213 

215 

217 

220 

222 

2.42 

224 

226 

228 

230 

232 

234 

237 

239 

241 

243 

2.43 

245 

247 

249 

251 

253 

255 

257 

259 

261 

264 

2.44 

266 

268 

270 

272 

274 

276 

278 

280 

282 

284 

2 

2.45 

286 

288 

290 

292 

294 

295 

297 

299 

301 

303 

2.46 

305 

307 

309 

311 

313 

315 

317 

319 

321 

323 

2.47 

324 

326 

328 

330 

332 

334 

336 

337 

339 

341 

2.48 

343 

345 

347 

349 

350 

352 

354 

356 

358 

359 

2.49 

361 

363 

365 

367 

368 

370 

372 

374 

375 

377 

2.5 

379 

396 

413 

430 

446 

461 

477 

492!   506 

520 

16 

2.6 

534 

547 

560 

573 

585 

59S 

609 

621 

632 

643 

12 

2.7 

653 

664 

674 

683 

693 

702 

711 

720 

728 

736 

9 

2.8  1 

744 

752 

760 

767 

774 

781 

788 

795 

801 

807 

7 

PROPORTIONAL  PARTS. 

A 

1 

2345 

6 

7 

9     9 

16 

1.6 

3.2    4.8    6.4    8.0 

9.6 

11.2 

12.8   14.4 

12 

1.2 

2.4   3.6   4.8   6.0 

7.2 

8.4 

9.6   10.8 

9 

0.9 

1.8   2.7   3.6   4.5 

5.4 

6.3 

7.2    8.1 

7 

0.7 

1.4   2.1    2.8    3.5 

4.2 

4.9 

5.6   6.3 

• 

VALUES    OF    NORMAL    PROBABILITY    INTEGRAL.      125 


TABLE  IV.— Continued. 


x/a 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

, 

2.9 

49813 

819 

825!  831 

836 

841 

846 

851 

856 

861 

5 

3.0 

865 

869 

873   878 

882 

886 

889 

893 

897 

900 

4 

3.1 

903 

906 

910 

913 

916 

918 

921 

924 

926 

929 

3 

3.2 

931 

934 

936 

938 

940 

942 

944 

946 

948 

950 

2 

3.3 

952 

953 

955 

957 

958 

960 

961 

962 

964 

965 

1 

3.4 

966 

968 

969 

970 

971 

972 

973 

974 

975 

976 

1 

3.5 

977 

978 

978 

979 

980 

981 

981 

982 

982 

983 

1 

3.6 

984 

985 

985 

986 

986 

987 

987 

988 

988 

989 

1 

3.7 

989 

990 

990 

990 

991 

991 

992 

992 

992 

992 

0 

3.8 

993 

993 

993 

994 

994 

994 

994   995 

995 

995 

0 

3.9 

995 

995 

996 

996 

996 

996 

996   996 

997 

997 

0 

4 

997 

998 

999 

999 

999 

000 

000 

000 

00. 

000 

0 

PROPORTIONAL,  PARTS. 

A 

1 

234 

5 

6 

789 

5 

0.5 

1.0    1.5    2.0 

2.5 

3.0 

3.5    4.0    4 

.5 

4 

0.4 

0.8    1.2    1.6 

2.0 

2.4 

2.8    3.2    3.6 

3 

0.3 

0.6    0.9    1.2 

1.5 

1.8 

2.1    2.4    2 

.7 

2 

0.2 

0.4    0.6    0.8 

1.0 

1.2 

1.4    1.6    1 

.8 

1 

0.1 

0.2    0.3    0.4 

0.5 

0.6 

0.7    0.8    0.9 

126  STATISTICAL  METHODS. 

V.— TABLE  OF  LOG  r  FUNCTIONS  OF  p  (see  pages  32-34). 


P 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.00 

97'50 

9500 

9251 

9003 

8755 

8509 

8263 

8017 

7773 

1.01 

"9!  997529 

7285 

7043 

6801 

6560 

0320 

6080 

5H41 

5602 

5365 

1.02 

5128 

4892 

4656 

4421 

4187 

3953 

3721 

3489 

3257 

3026 

1.03 

2796 

2567 

2338 

2110 

1883 

1056 

1430 

1205 

0981 

0757 

1.04 

0533 

0311 

0089 

9868 

§647 

§427 

§208 

§989 

grw 

§554 

1.05 

9.988338 

8122 

7907 

7692 

7478 

7265 

7052 

6841 

6629 

6419 

1.06 

6209 

6000 

5:91 

5583 

5376 

5169 

4903 

4758 

4553 

4349 

1.07 

4145 

3943 

3741 

3539 

3338 

3138 

2939 

2740 

2541 

2344 

1.08 

2147 

1951 

1755 

1560 

1365 

1172 

0978 

0786 

0594 

0403 

1.09 

0212 

0022 

§333 

§644 

§456 

§269 

§082 

8896 

8?  10 

8^25 

1.10 

9.978341 

8157 

7974 

7791 

7610 

7428 

7248 

7068 

6888 

6709 

1.11 

6531 

6354 

6177 

6000 

5825 

5650 

5475 

5301 

5128 

4955 

1.12 

4783 

4612 

4441 

4271 

4101 

3932 

3704 

3596 

3429 

3262 

1.13 

3096 

2931 

2766 

2602 

2438 

2275 

2113 

1951 

1790 

1029 

1.14 

1469 

1309 

1150 

0992 

0835 

0077 

0521 

0365 

0210 

0055 

1.15 

9.969901 

9747 

9594 

9442 

9290 

9139 

8988 

8838 

8688 

8539 

1.16 

8390 

8243 

8096 

7949 

7803 

7658 

7513 

7369 

7225 

7082 

1.17 

6939 

6797 

6655 

6514 

6374 

6234 

6095 

5957 

5818 

5081 

1.18 

5544 

5408 

5272 

5137 

5002 

4808 

4734 

4601 

4469 

4337 

1.19 

4205 

4075 

3944 

3815 

3686 

3557 

3429 

3302 

3175 

3048 

1.20 

2922 

2797 

2672 

2548 

2425 

2302 

2179 

2057 

1936 

1815 

1.21 

1695 

1575 

1456 

1337 

1219 

1101 

0984 

0867 

0751 

0636 

1.22 

0521 

0407 

0293 

0180 

0067 

9955 

8843 

9732 

9621 

9511 

1.23 

9.959401 

9292 

9184 

9076 

8968 

8801 

8755 

8019  . 

8544 

8439 

1.24 

8335 

8231 

8128 

8025 

7923 

7821 

7720 

7620 

7520 

7420 

1.25 

7321 

7223 

7125 

7027 

6930 

6S34 

6738 

6642 

6547 

6453 

1.26 

6359 

6267 

6173 

6081 

5989 

5898 

5807 

5716 

5627 

5537 

1.27 

5449 

5360 

5273 

5185 

5099 

5013 

4927 

4842 

4757 

4673 

1.28 

4589 

4506 

4423 

4341 

4259 

4178 

4097 

4017 

3938 

3858 

1.29 

3780 

3702 

3624 

3547 

3470 

3394 

3318 

3243 

3168 

3094 

1.30 

3020 

2947 

2874 

2802 

2730 

2659 

2588 

2518 

2448 

2379 

1.31 

2310 

2242 

2174 

2106 

2040 

1973 

1907 

1842 

1777 

1712 

1.32 

1648 

1585 

1522 

1459 

1397 

1336 

1275 

1214 

1154 

1094 

1.33 

1035 

0977 

0918 

0861 

0803 

0747 

0090 

0634 

0579 

0524 

1.34 

0470 

0416 

0362 

0309 

0257 

0205 

0153 

0102 

0051 

0001 

1.35 

9.949951 

9902 

9853 

9805 

9757 

9710 

9663 

9617 

9571 

9525 

1.36 

9430 

9435 

9391 

9348 

9304 

9202 

9219 

9178 

9136 

9095 

1.37 

9054 

9015 

8975 

8936 

8898 

8859 

8822 

8785 

8748 

8711 

1.33 

8676 

8640 

8005 

8571 

8537 

8503 

8470 

8437 

8405 

8373 

1.39 

8342 

8311 

8280 

8250. 

8221 

8192 

8163 

8135 

8107 

8080 

1.40 

8053 

8026 

8000 

7975 

7950 

7925 

7901 

7877 

7854 

7831 

1.41 

7808 

7786 

7765 

7744 

7723 

7703 

7683 

7064 

7645 

7626 

1.42 

7608 

7590 

7573 

7556 

7540 

7524 

7509 

7494 

7479 

7465 

1.43 

7451 

7438 

7425 

7413 

7401 

7389 

7378 

7368 

7358 

7348 

1.44 

7338 

7329 

7321 

7312 

7305 

7298 

7291 

7284 

7278 

7273 

1.45 

7268 

7263 

7259 

7255 

7251 

7248 

7246 

7244 

7242 

7241 

1.46 

7240 

7239 

7239 

7240 

7241 

7242 

7243 

7245 

7248 

7251 

1.47 

7254 

7258 

7262 

7266 

7271 

7277 

7282 

72>9 

7295 

7302 

1.48 

7310 

7317 

7326 

7334 

7343 

7353 

7303 

7373 

7384 

7395 

1.49 

7407 

7419 

7431 

7444 

7457 

7471 

7485 

7499 

7515 

7529 

TABLE   OF   LOG  r   FUNCTIONS.  127 

V.— TABLE  OF  LOG  r  FUNCTIONS  OP  p  (see  pages  32-34). 


p 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

.50 

9.947545 

7561 

7577 

7594 

7612 

7629 

7647 

7666 

76S5 

7704 

.51 

7724 

7744 

7764 

7785 

7806 

7828 

7850 

7873 

7896 

7919 

.52 

7943 

7967 

7991 

8016 

8041 

8067 

8093 

8120 

8146 

8174 

.53 

8201 

8229 

8258 

8287 

8316 

8346 

8376 

8406 

8437 

8468 

.54 

8500 

8532 

8564 

8597 

8630 

8664 

8698 

8732 

8767 

8802 

.55 

8837 

8873 

8910 

8946 

8983 

9021 

9059 

9097 

9135 

9174 

.56 

9214 

9254 

9294 

9334 

9375 

9417 

9458 

9500 

9543 

9586 

.57 

93  .'9 

9672 

9716 

9761 

9806 

9851 

9896 

9942 

9989 

6035 

.58 

9.950082 

0130 

0177 

0225 

0274 

0323 

0372 

0422 

0472 

0522 

.59 

0573 

0624 

0676 

0728 

0780 

0833 

0886 

0939 

0993 

1047 

.60 

1102 

1157 

1212 

1268 

1324 

1380 

1437 

1494 

1552 

1610 

.61 

1668 

1727 

1786 

1845 

1905 

1965 

2025 

2086 

2147 

2209 

.62 

2271 

2333 

2396 

2459 

2522 

2586 

2650 

2715 

2780 

2845 

.63 

2911 

2977 

3043 

3110 

3177 

3244 

3312 

3380 

3449 

3517 

.64 

3587 

3656 

3726 

3797 

3867 

3938 

4010 

4081 

4154 

4226 

.65 

4299 

4372 

4446 

4519 

4594 

4668 

4743 

4819 

4894 

4970 

.66 

5047 

5124 

5201 

5278 

5356 

5434 

5513 

5592 

5671 

5750 

.67 

5830 

5911 

5991 

6072 

6154 

6235 

6317 

6400 

6482 

6566 

.68 

6649 

6733 

6817 

6901 

6986 

7072 

7157 

7243 

7329 

7416 

.69 

7503 

7590 

7678 

7766 

7854 

7943 

8032 

8122 

8211 

8301 

.70 

8391 

8482 

8573 

8664 

8756 

8848 

8941 

9034 

9127 

9220 

.71 

9314 

9409 

9502 

9598 

9ii93 

9788 

9884 

9980 

6077 

6174 

.72 

9.960271 

0369 

0467 

0565 

0664 

0763 

0862 

0961 

1061 

1162 

.73 

1262 

1363 

1464 

1566 

1668 

1770 

1873 

1976 

2079 

2183 

.74 

2287 

2391 

2496 

2601 

2706 

2812 

2918 

3024 

3131 

3238 

.75 

3345 

3453 

3561 

3669 

3778 

3887 

3996 

4105 

4215 

4326 

.76 

4436 

4547 

4659 

4770 

4882 

4994 

5107 

5220 

5333 

5447 

.77 

5561 

5675 

5789 

5904 

6019 

6135 

6251 

6367 

6484 

6600 

.78 

6718 

6835 

6953 

7071 

7189 

7308 

7427 

7547 

7666 

7787 

.79 

7907 

8028 

8149 

8270 

8392 

8514 

8636 

8759 

8882 

9005 

.80 

9129 

9253 

9377 

9501 

9626 

9751 

9877 

5003 

5129 

6255 

.81 

9.970383 

0509 

0637 

0765 

0893 

1021 

1150 

1579 

1408 

1538 

.82 

1668 

1798 

1929 

2060 

2191 

2322 

2454 

2586 

2719 

2852 

.83 

2985 

3118 

3252 

3386 

3520 

3655 

3790 

3925 

4061 

4197 

.84 

4333 

4470 

4606 

4744 

4881 

5019 

5157 

5295 

5434 

5573 

.85 

5712 

5852 

5992 

6132 

6273 

6414 

6555 

6697 

6838 

6980 

.86 

7123 

7266 

7408 

7552 

7696 

7840 

7984 

8128 

8273 

8419 

.87 

8564 

8710 

8856 

9002 

9149 

9296 

9443 

9591 

9739 

9887 

.88 

9.980036 

0184 

0333 

0483 

0633 

0783 

0933 

1084 

1234 

1386 

.89 

1537 

1689 

1841 

1994 

2147 

2299 

2453 

2607 

2761 

2915 

.90 

3069 

3224 

3379 

3535 

3690 

3846 

4003 

4159 

4316 

4474 

.91 

4631 

4789 

4947 

5105 

5264 

5423 

5582 

5742 

5902 

6062 

.92 

6223 

6383 

6544 

6706 

6867 

7029 

7192 

7354 

7517 

7680 

.93 

7844 

8007 

8171 

8336 

8500 

8665 

8830 

8996 

9161 

9327 

.94 

9494 

9660 

9827 

9995 

6162 

5330 

5498 

6666 

5835 

1004 

1.95 

9.991173 

1343 

1512 

1683 

1853 

2024 

2195 

2306 

2537 

2709 

1.96 

2881 

3054 

3227 

3399 

3573 

3746 

3920 

4094 

4269 

4443 

1.97 

4618 

4794 

4969 

5145 

5321 

5498 

5674 

5851 

6029 

6206 

1.98 
1.99 

6384 
8178 

6562 
8359 

6740 
8540 

6919 

8722 

7098 
8903 

7277 
9085 

7457 
9268 

7637 
9450 

7817 
9633 

7997 
9816 

128 


STATISTICAL   METHODS. 


VI.— TABLE  OF  REDUCTION  FROM  COMMON  TO  METRIC  SYSTEM.   *, 


Inches  to  Millimeters. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

25.40 

50. 

RO 

76.20 

101.60 

127.00 

152.40 

177 

.80 

203.20 

228.60 

10 

279.40 

304. 

SO 

330.19 

355.59 

380.99 

406.39 

431 

.79 

457.19 

482.59 

?0 

533.39 

558. 

79 

584.19 

6 

09.59 

634.99 

660.  3^ 

685 

.79 

7 

1.19 

736.59 

30 

787.39 

812. 

79 

838  19 

863.59 

888.99 

914.39 

939 

.78 

965.18 

990.58 

40 

1041.4 

1066. 

8 

1092.2 

1117.6 

1143.0 

1168.4 

1193 

.8 

1219.2 

1244.6 

50 

1295.4 

1320. 

R 

1346.2 

13 

71.6 

1397.0 

1422.4 

1447 

.8 

14 

"3.2 

1498.6 

60 

1549.4 

1574. 

R 

1600.2 

1625.6 

1651.0 

1076.4 

1701 

.8 

1727.2 

1 

752.6 

70 

1803.4 

1828. 

R 

1854.2 

1879.6 

1905.0 

1930.4 

1955 

.8 

1981.2 

2006.6 

80 

2057.4 

2082. 

8 

2108.2 

21 

33.6 

2159.0 

2184.4 

2209 

.8 

22 

35.2 

2 

260.6 

90 

2311.4 

2336.8 

2362.2 

2387.6 

2413.0 

2438.4 

2463 

.8 

2489.2 

2514.6 

Twelfths. 

Sixteenths. 

1/12 

2/12 
3/12 
4/12 

2.12 
4.23 
6.35 
8.47 

7/12 
8/12 
9/12 
10/12 

14.82 
16.93 
19.05 
21.17 

1/16 

1/8 
3/16 

1/4 

1.59 
3.17 
4.76 
6.35 

5/16 

3/8 
7/16 
1/2 

7.94 
9.52 
11.11 

12.70 

9/16 

5/8 
11/16 

3/4 

14.29 

15.87 
17.46 
19.05 

13/16 

7/8 
15/16 
1 

20.64 
22.22 
23.81 

25.40 

5/12 

10.58 

11/12 

23.28 

6/12 

12.70 

12/12 

25.40 

TABLE  VII— MINUTES  AND  SECONDS  IN  DECIMALS  OF  A  DEGREE. 


' 

o 

' 

0 

' 

0 

- 

o 

" 

o 

" 

o 

1 

.016666 

21 

.  350000 

41 

.683333 

1 

.000278* 

21 

.005833 

41 

.011389 

2 

.033333 

22 

.  366666 

42 

.700000 

2 

.000556 

22 

.006111 

42 

.011667 

3 

.050000 

23 

.383333 

43 

.716666 

3 

.000833 

23 

.006389 

43 

.011944 

4 

.066666 

24 

.400000 

44 

.733333 

4 

.001111 

24 

.006667 

44 

.012222 

5 

.083333 

25 

.416666 

45 

.750000 

5 

.001389 

25 

.006944 

45 

.012500 

6 

.  100000 

26 

.433333 

46 

.766666 

6 

.001667 

26 

.007222 

46 

.012778 

7 

.116666 

27 

.  450000 

47 

.783333 

7 

.001944 

27 

.007500 

47 

.013056 

8 

.133333 

28 

.466666 

48 

.800000 

8 

.002222 

28 

.007778 

48 

.013333 

9 

.  150000 

29 

.483333  49 

.816666 

9 

.  002500 

29 

.008056 

49 

.013611 

10 

.  166666 

30 

.  500000 

50 

.833333 

10 

.002778 

30 

.008333 

50 

.013889 

11 

.  183333 

31 

.516666 

51 

.850000 

11 

.  003056 

31 

.008611 

si 

.014167 

12 

.  200000 

32 

.  533333 

52 

.866666 

12 

.  003333 

32 

.008889 

52 

.014444 

13 

.216666 

33 

.  550000 

53 

.883333 

13 

.003611 

33 

.009167 

53 

.014722 

14 

.  233333 

34 

.  566666 

54 

.  900000 

14 

.  003889 

34 

.  009444 

54 

.015000 

15 

.  250000 

35 

.583333 

55 

.916666 

15 

.004167 

35 

.009722 

55 

.015278 

16 

.  266666 

36 

.600000 

56 

.933333 

16 

.004444 

36 

.010000 

56 

.015556 

17 

.  283333 

37 

.616666 

57 

.950000 

17 

.004722 

37 

.010278 

57 

.015833 

18  .300000 

38 

.  633333 

58 

.966666 

18 

.  005000 

38 

.010556 

58 

.016111 

19 

.316666 

39 

.650000 

59 

.983333 

19 

.005278 

39 

.010833 

59 

.016389 

20 

.333333 

40 

.666666 

60 

1.000000 

20 

.005556 

40 

.011111 

60 

.016667 

*  .0002777778. 


FIRST   TO   SIXTH   POWERS   OF   INTEGERS. 


129 


TABLE  VIII.— FIRST  TO  SIXTH  POWERS  OF  INTEGERS  FROM  1  TO  50. 


Powers. 

First. 

Second. 

Third. 

Fourth. 

Fifth. 

Sixth. 

1 

1 

1 

1 

1 

1 

2 

4 

8 

16 

32 

64 

3 

9 

27 

81 

243 

729 

4 

16 

64 

256 

1024 

4096 

5 

25 

125 

625 

3125 

15625 

6 

36 

216 

1296 

7776 

46656 

7 

49 

343 

2401 

16807 

117649 

8 

64 

512 

4096 

32768 

202  144 

9 

81 

729 

6561 

59049 

531441 

10 

100 

1000 

10000 

100000 

1000000 

11 

121 

1331 

14641 

161051 

1771561 

12 

144 

1728 

20736 

248832 

2985984 

13 

169 

2197 

28561 

371293 

4826809 

14 

196 

2744 

38416 

5378-24 

7529536 

15 

225 

3375 

50625 

759375 

11390625 

16 

256 

4096 

65536 

1048576 

16777216 

17 

289 

4913 

83521 

1419857 

24137569 

18 

324 

5832 

104976 

1889568 

34012-J24 

19 

361 

6859 

180821 

2476099 

47045881 

20 

400 

8000 

160000 

3200000 

64000000 

21 

441 

9261 

194481 

4084101 

85766121 

22 

484 

10648 

234256 

5153632 

113379904 

23 

529 

12167 

279841 

643U343 

148035889 

24 

576 

13824 

331776 

7962624 

191102976 

25 

625 

156','5 

390625 

9765625 

244140025 

26 

676 

17576 

456976 

11881376 

308915776 

27 

7*9 

19683 

531441 

14348907 

387420489 

28 

784 

21952 

614656 

17210368 

481890304 

29 

841 

24389 

707281 

30511149 

594823321 

30 

900 

27000 

810000 

24300000 

729900000 

31 

961 

29791 

923521 

28629151 

887503681 

32 

1024 

32768 

1048576 

33554432 

1078741824 

33 

1089 

35937 

1185921 

39I35W3 

1:*)  14(57%'.) 

34 

1156 

39304 

1336336 

45435424 

1544SW4N5 

35 

1225 

42875 

1500625 

52521875 

1S3SXJ65625 

36 

1296 

46656 

1679616 

60464176 

2176782336 

37 

1369 

50653 

1874161 

88648967 

2565726409 

38 

1444 

54872 

2085136 

79886168 

30KMM53S4 

39 

1521 

59319 

2313441 

90224199 

351S7437J5I 

40 

1600 

64000 

2560000 

102400000 

4IM5000000 

41 

1681 

68921 

2825761 

ItJWOBOJ 

4750104241 

42 

1764 

74088 

3111696 

13(H5912U2 

54*9081744 

43 

1849 

79507 

3418801 

147008448 

15321  3»KW«.» 

44 

1936 

85184 

3748096 

164916884 

725631 

45 

202.") 

91125 

4100625 

1H4528125 

KHH765625 

46 

2116 

97336 

4477456 

:>o:><x>-»i:<; 

9471:.' 

47 

2209 

103823 

4S796H1 

8292)49007 

1077»ttl5888 

4S 
49 
50 

2304 
2401 
2500 

110592 
117649 
125000 

5WS416 
57154801 
6250000 

864808968 

282475249 

312500000 

12230590464 

13S4I2S72W 
15T,:.'50l)0000 

130 


STATISTICAL  METHODS. 


TABLE  IX.— PROBABLE  ERRORS  OF  THE  COEFFICIENT  OF  COR- 
RELATION  FOR  VARIOUS  NUMBERS  OF  OBSERVATIONS  OR 
VARIATES  (n)  AND  FOR  VARIOUS  VALUES  OF  r. 

Decimal  point,  properly  preceding  each  entry,  is  omitted.  (Specially  Cal- 
„._,  culated.) 


Number 

Correlation  Coefficient  r. 

of  Obser- 

vations- 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

20 

1508 

1493 

1448 

1373 

1267 

1131 

0965 

30 

1231 

1219 

1182 

1121 

1035 

0924 

0788 

40 

1067 

1056 

1024 

0971 

0896 

0800 

0683  . 

50 

0954 

0944 

0915 

0868 

0801 

0715 

0610 

60 

0871 

0862 

0836 

0793 

0731 

0653 

0557 

70 

0806 

0798 

0774 

0734 

0677 

0605 

0516 

80 

0754 

0747 

0724 

0686 

0633 

0566 

0483 

90 

0711 

0704 

0683 

0647 

0597 

0533 

0455 

100 

0674 

0668 

0648 

0614 

0567 

0506 

0432 

150 

0551 

0546 

0529 

0501 

0463 

0413 

0352 

200 

0477 

0472 

0458 

0434 

0401 

0358 

0305 

250 

0426 

0421 

0409 

0387 

0358 

0319 

0272 

300 

0389 

0386 

0374 

0354 

0327 

0292 

0249 

400 

0337 

0334 

0324 

0307 

0283 

0253 

0216 

500 

0302 

0299 

0290 

0274 

0253 

0226 

0193 

600 

0275 

0272 

0264 

0251 

0232 

0207 

0176 

700 

0255 

0252 

0245 

0232 

0214 

0191 

0163 

800 

0239 

0236 

0229 

0217 

0200 

0179 

0153 

900 

0225 

0222 

0216 

0205 

0189 

0169 

0144 

1000 

0213 

0211 

0205 

0194 

0179 

0160 

0137 

2000 

0151 

0149 

0145 

0137 

0127 

0113 

0097 

5000 

0095 

0094 

0092 

0087 

0080 

0072 

0061 

0.65 

0.7 

0.75 

0.8 

0.85 

0.9 

0.95 

20 

0871 

0769 

0660 

0543 

0419 

0287 

0147 

30 

0711 

0628 

0539 

0444 

0342 

0234 

0120 

40 

0616 

0544 

0467 

0384 

0296 

0203 

0104 

50 

0551 

0486 

0417 

0343 

0265 

0181 

0093 

60 

0503 

0444 

0381 

0313 

0241 

0165 

0085 

70 

0466 

0411 

0353 

0290 

0224 

0153 

0079 

80 

0436 

0385 

0330 

0271 

0209 

0143 

0074 

90 

0411 

0363 

0311 

0256 

0197 

0135 

0069 

100 

0391 

0345 

0294 

0242 

0187 

0128 

0066 

150 

0318 

0281 

0241 

0198 

0153 

0105 

0054 

200 

0275 

0243 

0209 

0172 

0133 

•0091 

0047 

250 

0246 

0218 

0187 

0154. 

0118 

0081 

0042 

300 

0225 

0199 

0170 

0140 

0108 

0074 

0038 

400 

0195 

0172 

0148 

0122 

0094 

0064 

0033 

500 

0174 

0154 

0132 

0109 

0084 

0057 

0029 

600 

0159 

0140 

0121 

0099 

0076 

0052 

0027 

700 

0147 

0130 

0112 

0092 

0071 

0049 

0025 

800 

0138 

0122 

0105 

0086 

0066 

0045 

0023 

900 

0130 

0114 

0098 

0081 

0062 

0043 

0022 

1000 

0123 

0109 

0093 

0077 

0059 

0041 

0021 

2000 

0087 

0077 

0066 

0054 

0042 

0029 

0014 

5000 

0055 

0049 

0042 

0034 

0026 

0018 

0009 

TABLE   X.— SQUARES,    CUBES,    ETC. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

"  "  —"I 
Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1  .  2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.2360680 

1.7099759 

.200000000 

6 

36 

216 

2  4494897 

1.8171206 

.166600667 

7 

49 

343 

2.6457513 

1.9129312 

.  142857143 

8 

64 

512 

2.8284271 

2.0000000 

.125000000 

9 

81 

729 

3.0000000 

2.0800837 

.111111111 

10 

100 

1000 

3.1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248 

2.2239801 

.090909091 

12 

144 

1728 

3.4641016 

2.2894286 

.083333333 

13 

169 

2197 

3.6055513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.071428571 

15 

225 

3375 

3.8729833 

2.4662121 

.000000007 

16 

256 

4096 

4.0000000 

8.5196401 

.002500000 

17 

289 

4913 

4.1231056 

2.5712816 

.05882:1529 

18 

324 

5832 

4.3426407 

2.6207414 

.055555556 

19 

361 

0859 

4.3588989 

2.6684016 

.0520:31579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

1)261 

4.5825?T>7 

2.7589243 

.047019048 

22 

484 

10648 

4.6904158 

2.8020393 

.045454545 

23 

529 

12167 

4.7958315 

2.8438670 

.043478261 

24 

576 

13824 

4.8989795 

2.8844991 

.041000007 

25 

025 

15625 

5.0000000 

2.1)240177 

.040000000 

26 

676 

17576 

5.0990195 

2.9624960 

.038401538 

27 

729 

19683 

5.1961524 

3.0000000 

.037037037 

28 

784 

21952 

5.2915026 

8.0865889 

.035714286 

29 

841 

24389 

5.3851648 

3.0723168 

.034482759 

30 

900 

27000 

5.4772256 

3.1072325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.032258005 

32 

1024 

32768 

5.6568542 

3.1748021 

.031250000 

33 

1089 

35937 

5.7445626 

3.2075343 

030303030 

34 

1156 

39304 

5.8309519 

3.2396118 

.020411765 

35 

1225 

42875 

5.9160798 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

38 

1444 

54872 

0.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449980 

3.3912114 

.025041020 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1681 

68921 

6.4031242 

3.4482172 

.(121:590344 

42 

17G4 

74088 

0.4807407 

3  4760266 

.023809524 

43 

184!) 

79507 

6.5574385 

3.5033981 

.023255814 

44 

1936 

85184 

6.0332496 

3.5303483 

.022727273 

45 

2025 

91125 

6.7082039 

3.5568933 

.022222222 

46 

2116 

07336 

6.7823300 

3.5830479 

.021739130 

47 

2209 

103823 

6.8556546 

3.6088261 

.(121276600 

48 

2304 

110592 

6.  92820:  W 

3.6342411 

.(M:s:«333 

49 

2401 

I  117649 

7.0000000 

3.0593057 

.020408163 

50 

2500 

125000 

7.0710678 

3.6840314 

.020000000 

51 

2G01 

132651 

7.1414284 

3.7084298 

.010(507843 

52 

2704 

140608 

7.2111026 

3.7325111 

.019330769 

68 

2809 

148877 

7.2801099 

8.7662858 

,018607935 

54 

2916 

157464 

7.3484(592 

3.7797631 

.018518519 

55 

3025 

166375 

7.4161985 

8.8009535 

.018181818 

56 

3136 

1?'5616 

7.4833148 

8.8856634 

.017857148 

57 

3249 

185193 

7.549S::u 

8.8485011 

58 

3304 

105112 

7.6157781 

3  8708766 

.017241879 

59 

3481 

206379 

7.6811457 

8.8999965 

.016940153 

60 

3600 

21BOOO 

7.7459667 

3.0148676 

.01  6666(  :»  .7 

61 

3721 

226981 

7.8102497 

3.0.304072 

.0163931  »:J 

63 

3844 

238838 

7.8740079 

3.0578915 

.010120032 

131 


TABLE   X. — SQUARES,   CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

63 

3969 

250047 

7.9372539 

3.9790571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015625000 

65 

4225 

274625 

8.0622577 

4.0207256 

.015384615 

66 

4356 

287496 

8.1240384 

4.0412401 

.015151515 

67 

4489 

300763 

8.1853528 

4.0615480 

.014925373 

68 

4624 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70 

4900 

343000 

8.3666003 

4.1212853 

.014285714 

71 

5041 

357911 

8.4261498 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4.1601676 

.013888889 

73 

5329 

389017 

8.5440037 

4.1793390 

013698630 

74 

5476 

405224 

8.6023253 

4.1983364 

.013513514 

75 

5625 

421875 

8.6602540 

4.2171633 

.013333333 

76 

5776 

438976 

8.7177979 

4.2358236 

.013157895 

77 

5929 

456533 

8.7749644 

4.2543210 

.012987013 

78 

6084 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012658228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

6561 

531441 

9.0000000 

4.3267487 

.012345679 

83 

6724 

55136S 

9.0553851 

4.3444815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3620707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904762 

85 

7225 

614125 

9.2195445 

4.3968296 

.011764706 

86 

7396 

636056 

9.2736185 

4.4140049 

.011627907 

87 

7569 

658503 

9.3273791 

4.4310476 

.011494253 

88 

7744 

681472 

9.3808315 

4.4479602 

.011363636 

89 

7921 

704969 

9.4339811 

4.4647451 

.011235955 

90 

8100 

729000 

9.4868330 

4.4814047 

.011111111 

91 

8281 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778688 

9.5916G30 

4.5143574 

.010869565 

93 

8649 

804357 

9.643G508 

4.5306549 

.010752688 

94 

8836 

830584 

9.6953597 

4.5468359 

.010638298 

95 

9025 

857375 

9.7467943 

4.5629026 

.010526316 

96 

9216 

884736 

9.7979590 

4.5788570 

.010416667 

97 

9409 

912673 

9.8488578 

4.5947009 

.010309278 

98 

9604 

941192 

9.8994949 

4.6104363 

.010204082 

99 

9801 

970299 

9.9498744 

4.6260650 

.010101010 

100 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.C570095 

.009900990 

102 

10404 

1061208 

10.0995049 

4.6723287 

.009803922 

103 

10609 

1092727 

10.1488916 

4.G875482 

.009708738 

104 

10816 

1124864 

10.1980390 

4.7026694 

.009615385 

105 

11025 

1157625 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

10.2956301 

4.7326235 

.009433962 

107 

11449 

1225043 

10.3440804 

4.7474594 

.009345794 

108 

11664 

1259712 

10.3923048 

4.7622032 

.00)259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

.009174312 

110 

12100 

1331000 

10.4880885 

4.7914199 

.009090909 

111 

12321 

1307631 

10.535C538 

4.8058955 

.009009009 

112 

12544 

1404928 

10.5830052 

4.8202845 

.0089285:1 

113 

12769 

1442897 

10.6301458 

4.8345881 

.008849558 

114 

12996 

1481544 

10.(.7707&3 

4.8488076 

.008771930 

115 

13225 

152087'5 

10.7238053 

4.8629442 

.008095652 

116 

13456 

1560896 

10.7703296 

4.8769990 

.008020690 

117 

13689 

1601613 

10.8166538 

4.8909732 

.008547009 

118 

13924 

1643032 

10.8627805     4.9048681 

.008474576 

119 

14161 

1685159 

10.9087121 

4.9186847 

.008403361 

120 

14400 

1728000 

10.9544512 

4.9324342 

.008333333 

121 

14641 

1771561 

11.00  0000 

4.9460874 

.0082G4463 

1S2 

14884 

1815848 

11.0453610 

4.9596757 

.008196721 

123 

15129 

1860867 

11.0905365 

4.9731898 

.008130081 

124 

15376 

1906624 

11.1355287 

4.98G6310 

.008064516 

132 


CUBE    ROOTS,    AtfD    RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

125 

15625 

1953125 

11.1803399 

5.0000000 

.008000000 

126 

15876 

2000376 

11.2249722 

5.0132979 

.007936508 

127 

16129 

2048383 

11.2694277 

5.0265257 

.007874016 

128 

16384 

2097152 

11.3137085 

5.0396842 

.007812500 

128 

16641 

2146689 

11.3578167 

5.0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131 

i;iei 

22480U1 

11.4455231 

6.0787531 

.007633588 

132 

17424 

2299968 

11.4891253 

5.0916434 

.007575758 

133 

17689 

2352637 

11.5325626 

5.1044687 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

.007407407 

136 

18496 

2515456 

11.6619038 

5.1425632 

.007352941 

137 

18769 

2571353 

11.7046999 

5.1551367 

.007299270 

138 

19044 

2628072 

11.7473401 

5.1676493 

.007246377 

139 

19321 

2685619 

11.7898261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

5.1924941 

.007142857 

141 

19881 

2803221 

11.8743421 

5.2048279 

.007092199 

142 

2Q104 

2863288 

11.9163753 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2985984 

12.0000000 

5.2414828 

.006944444 

145 

21025 

3048625 

12.0415946 

5.2535879 

.006890552 

146 

21316 

3112136 

12.0830460 

5.2656374 

.006849315 

147 

21009 

3176523 

12.1243557 

5.2776321 

.006802721 

148 

21904 

3241792 

12.1655251 

5.2895725 

.006756757 

149 

22201 

3307949 

12.2065556 

5.3014592 

.006711409 

150 

22500 

3375000 

12.^474487 

5.3132928 

.006666667 

151 

22801 

3442951 

12.2882057 

5.3250740 

.006622517 

152 

23104 

3511808 

12.3288280 

5.3368033 

.006578947 

153 

23409 

3581577 

12.3693169 

5.3484812 

.006535948 

154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

155 

24025 

37238?o 

12.4498996 

5.3716854 

.006451613 

156 

24336 

3796416 

12.4899960 

5.  3832126 

.006410256 

157 

24649 

3869893 

12.5299641 

5  3946907 

.006369427 

158 

24964 

3944312 

12.5698051 

5.4061202 

.006329114 

159 

25281 

4019679 

12.6095202 

5.4175015 

.006289308 

160 

25600 

4096000 

12.6491106 

5.4288352 

.006250000 

161 

25921 

4173281 

12.6885775 

5.4401218 

.006211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172840 

163 

26569 

4330747 

12.7671453 

5.4625556 

.006134909 

164 

26896 

4410944 

12.8062485 

5.4737037 

.006097561 

165 

27225 

4492125 

12.8452326 

5.4848066 

.006060606 

166 

27556 

4574296 

12.8840987 

5.4958647 

.006024096 

167 

27889 

4657463 

12  9228480 

5.5068784 

.005988024 

168 

28224 

4741632 

12.9614814 

5.5178484 

.005952381 

169 

28561 

4826809 

13.0000000 

5.5287748 

.005917160 

17'0 

28900 

4913000 

13.03S4048 

5.5396583 

.00588235:5 

171 

29241 

5000211 

13.0766968 

5.5504991 

.  005847  •'.):>:* 

172 

29584 

5088448 

13.1148770 

5.5612978 

.005813953 

173 

29929 

5177717 

13.1529464 

5.5720546 

.005780347 

174 

30276 

5268024 

13  1909060 

5.5827702 

.00574712(5 

175 

30625 

5359375 

13  2287566 

5.5934447 

.0057142SU 

176 

30976 

5451776 

13.2664992 

5.60407S7 

.005681818 

177 

31329 

5545233 

13.3041347 

5.6146724 

.005649718 

178 

31684 

5639752 

13  3410041 

6.6252263 

.005617978 

179 

32041 

5735339 

13.3790882 

5.6357408 

.005586592 

180 

32400 

5&32000 

13.4164079 

5.6462162 

.005.V 

181 

82761 

59297'41 

13.4536240 

5.6566528 

.005524H*  tt 

182 

33124 

6028568 

13.4907376 

5.6670511 

.0054!)}:."5 

183 

33489 

6128487 

13.5277493 

5  6774114 

.0054011S1 

184 

33856 

6229504 

13.5646600 

5.6877340 

.006481788 

185 

34225 

6331625 

13.6014705 

5.69801!»2 

.0054054it:> 

186 

34596 

6434856 

13.6381817 

6.7082675 

.005376344 

133 


TABLE   X.  -SQUARES,   CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

5.7286543 

.005319149 

189 

35721 

6751269 

13.7477271 

5.7387936 

.005291005 

190 

36100 

6859000 

13.7840488 

5.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

5.7589652 

.005235602 

192 

36864 

7077888 

13.8564065 

5.76891)82 

.005208333 

193 

37249 

7189057 

13.8924440 

5.7789966 

.005181347 

194 

37636 

7301384 

13.9283883 

5.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

5.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

5.8087857 

.005102041 

397 

38809 

7645373 

14.0356688 

5.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

5.8284767 

.005030505 

199 

39601 

7880599 

14.1067360 

5.8382725 

.005025126 

200 

40000 

8000000 

14.1421356 

5.8480355 

.005000000 

201 

40401 

8120601 

14.1774469 

5.8577660 

.004975124 

202 

40804 

8242408 

14.2126704 

5.8674643 

.004950495 

203 

41209 

8365427 

14.2478068 

5.8771307 

.004926108 

204 

41616 

8489664 

14.2828569 

5.8867653 

.004901961 

205 

42025 

8615125 

14.3178211 

5.8963685 

.004878049 

206 

42436 

8741816 

14.3527001 

5.9059406 

.004854369 

207 

42849 

8869743 

14.3874946 

5.9154817 

.0048,°,0918 

208 

43264 

8998912 

14.4222051 

5.9249921 

.004807692 

209 

43681 

9129329 

14.4568323 

5.9344721 

.004784689 

210 

44100 

9261000 

14.4913767 

5.9439220 

.004761905 

211 

44521 

9393931 

14.5258390 

5.9533418 

.004739336 

212 

44944 

9528128 

14.5602198 

5.9627320 

.004716981 

213 

45369 

9663597 

14.5945195 

5.9720926 

.004694836 

214 

45796 

9800344 

14.6287388 

5.9814240 

.004672897 

215 

46225 

9938875 

14.6628783 

5.9907264 

.004651163 

216 

46656 

10077696 

14.6969385 

6.0000000 

.004629630 

217 

47089 

10218313 

14.7309199 

6.0092450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986486 

6.0276502 

.004566210 

220 

48400 

10648000 

14.8323970 

6.0368107 

.004545455 

221 

48841 

10793861 

14.8660687 

6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6.0550489 

.004504505 

223 

49729 

11089567 

14.9331845 

6.0641270 

.004484306 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

15.0000000 

6.0822020 

.004444444 

226 

51076 

11543176 

15.0332964 

6.0911994 

.004424779 

227 

51529 

11697083 

15.0665192 

6.1001702 

.004405286 

228 

51984 

11852352 

15.0996689 

6.1091147 

.004385965 

229 

52441 

12008989 

15.1327460 

6.1180332 

.004366812 

230 

52900 

12167000 

15.1657509 

6.1269257 

.004347826 

251 

53361 

12326391 

15.1986842 

6.1357924 

.004329004 

232 

53824 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

54289 

12649337 

15.2643375 

6.1534495 

.004291845 

234 

54756 

12812904 

15.2970585 

6.1622401 

.004273504 

235 

55225 

12977875. 

15.3297097 

6.1710058 

.004255319 

236 

55696 

13144256- 

15.3622915 

6.1797466 

.004237288 

237 

56169 

13312053 

15.3948043  !   6.1884628 

.004219409 

238 

56644 

13481272 

15.4272486 

6.1971544 

.004201681 

239 

57121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

15  4919334 

6.2144650 

.004166667 

241 

58081 

13997521 

15.5241747 

6.2230843 

.004149378 

242 

58564 

14172488 

15.5563492 

6.2316797 

.004132231 

243 

59049 

14348907 

15.5884573 

6.2402515 

.004115226 

244 

59536 

14526784 

15.6204994 

6.2487998 

.004098561 

245 

60025 

14706125 

15.6524758 

6.2573248 

.004081633 

246 

60516 

14886936 

15.6843871 

6.2658266 

.004065041 

247 

61009 

15069223 

15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992 

15.7480157 

6.2827613 

.004032258 

134 


CUBE    ROOTS,    AND    RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

249 

62001 

15438249 

15.7797338 

6.2911946 

.004010064 

250 

62500 

15625000 

15.8113883 

6.29960.53 

.004000000 

251 

0300  1 

15813251 

15.8429795 

6.307! 

.003lfc$4U04 

252 

63504 

16003008 

15.8745079 

6.31035% 

.008968254 

253 

64009 

16194277 

15.9059737 

6.3247035 

.00.7.): 

254 

64516 

16387064 

15.9373775 

6.3330256 

.003937008 

255 

65025 

16581375 

15.9087194 

6.3413257 

.003921509 

256 

65536 

16777216 

16.0000000 

6.3490042 

.003900250 

257 

66049 

16974593 

16.0312195 

6.3578011 

.  003891  U51 

258 

66564 

17173512 

16.0023784 

6.3600908 

.003875909 

259 

67081 

17373979 

16.0934769 

6.  3743111 

.003801004 

260 

67600 

17576000 

16.1245155 

6.3825043 

.003846154 

261 

68121 

17779581 

16.1554944 

0.31)00705 

.003831418 

262 

68644 

17984723 

16.1864141 

6.3988279 

.003810794 

263 

69169 

18191447 

16.2172747 

6.4009585 

.003802281 

264 

69696 

18399744 

16.2480768 

6.4150687 

.003787879 

265 

70225  ' 

18609625 

16.2788206 

6.4231583 

.0037; 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759398 

267 

71289 

19034163 

16.3401346 

6.43927'07 

.003745318 

268 

71824 

19248332 

16.3707055 

6.4473057 

.003731343 

269 

72361 

19465109 

16.4012195 

6.4553148 

.003717472 

270 

72900 

19683000 

16.4316767 

6.4033041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003090037 

272 

7'398t 

20123648 

16.4924225 

6.4792236 

.00307-0471 

273 

74529 

20346417 

16.5227116 

6.4871541 

.00366^3004 

274 

75076 

20570824 

16.5529454 

6.4950653 

.003049035 

275 

75625 

20796875 

16.5831240 

6.5029572 

.003031)304 

276 

76176 

21024576 

16.6132477 

6.5108300 

.003623188 

277 

76729 

21253933 

16.6433170 

6.5186839 

.0013010108 

278 

77284 

21484952 

16.0733320 

6.5265189 

.003597122 

279 

77841 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

6.5421326 

.003571429 

281 

78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425768 

16.7928556 

6.5576722 

.003540099 

283 

80089 

22665187 

16.8220038 

6.5654144 

.003533509 

284 

80656 

22906304 

16.8522995 

6.5731385 

.003521127 

285 

81225 

23149125 

16.8819430 

6.5808443 

.003508772 

286 

81796 

23393656 

16.9115345 

6.5885323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.5902023 

.00348  4:{\!1 

288 

82944 

23887872 

16.9705027 

6,6088545 

.003t; 

289 

83521 

24137509 

17.0000000 

G.  01  14690 

.003  100208 

290 

84100 

24389000 

17.0293864 

6.6191060 

.003418270 

291 

84681 

24042171 

1  .0587221 

6.  02071).-,  l 

.008486426 

292 

&5204 

24897088 

1  .0880075 

6.63I2S1 

.006424658 

293 

85849 

25153757 

1  .1172428 

6.6418522 

.003412%'.) 

294 

86433 

25412184 

1  .1464282 

6.6493998 

.00340i:;i;i 

295 

87025 

256?'237'5 

1  .1755640 

6.6509302 

.0033S!)>:;i 

296 

87616 

25931336 

1  .2046505 

6.6644437 

1  00337* 

297 

88209 

26198073 

1  .2336879 

6.6719403 

.«>n:::507003 

298 

8880  1 

26463:.!*  3 

1  .2620705 

6.6794200 

.0038! 

299 

89401 

26730899 

1  .2910105 

C.G808831 

.003.-J1 

300 

90000 

27000000 

1  .3205081 

6  crm-m") 

.00888 

301 

90601 

27270901 

1  .3493516 

6.7017588 

.008888859 

302 

91204 

27'543008 

1  .3781472 

6.7001789 

;  i  -j:,s 

303 

91809 

27818127 

1  .4008952 

6.7165700 

304 

92416 

2809  41(51 

1  .4:355958 

6.72395US 

s'i;  i 

305 

93025 

28372625 

1  .4642492 

6.7818159 

.00327* 

306 

93636 

28652616 

1  .4928557 

6.7:386641 

.  003207  !«;t 

307 

94249 

28934443 

1  .5214155 

6.7459967 

308 

94864 

29218112 

1  .54992SS 

0.7533134 

46758 

309 

95481 

295031  i--".) 

1  .57K5'.r,S 

G.  70001  i:5 

8246 

310 

96100 

29791000 

i  .0008109 

0.7078995 

.003225806 

TABLE   X. — SQUARES,    CUBES,   SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Kuots. 

Cube  Roots. 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

312 

97344 

30371328 

17.6035217 

6.7824229 

.003205128 

313 

97969 

30664297 

17.6918060 

6.7896613 

.003194888 

314 

98596 

30959144 

17.7200451 

6.7968844 

.003184713 

315 

99225 

31255875 

17.7482393 

6.8040921 

.003174603 

316 

99856 

31554496 

17.7763888 

6.8112847 

.003164557 

317 

100489 

31855013 

17.8044938 

6.8184620 

.003154574 

318 

101124 

32157432 

17.8325545 

6.8256242 

.003144654 

319 

101761 

32461759 

17.8605711 

6.8327714 

.003134796 

320 

102400 

32768000 

17.8885438 

6.8399037 

.003125000 

321 

103041 

33076161 

17.9164729 

6.8470213 

.003115205 

322 

103684 

33386248 

17.9443584 

6.8541240 

.003105590 

323 

104329 

33698267 

17.9722008 

6.8612120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8682855 

.003086420 

325 

105625 

34328125 

18.0277564 

6.8753443 

.003076923 

320 

106276 

34645976 

18.0554701 

6.8823888 

.003067485 

327 

106929 

34965783 

18.0831413 

6.8894188 

.003058104 

328 

107584 

35287552 

18.1107703 

6.8964345 

.003048780 

o29 

108241 

35611289 

18.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

6.9104232 

.003030303 

331 

109561 

36264691 

18.1934054 

6.9173964 

.003021148 

332 

110224 

36594368 

18.2208672 

6.9243556 

.003012048 

333 

110889 

36926037 

18.2482876 

6.9313008 

.003003003 

334 

111556 

37259704 

18.2756669 

6.9382321 

.002994012 

335 

112225 

37595375 

18.3030052 

6.9451496 

.002985075 

330 

112896 

37933056 

18.3303028 

6.9520533 

.002976190 

337 

113569 

38272753 

18.3575598 

6.9589434 

.002967359 

338 

114244 

38614472 

18.3847763 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

.002949853 

340 

115600 

39304000 

18.4390889 

6.9795321 

.002941176 

341 

116281 

39651821 

18.4661853 

6.9863681 

.002932551 

342 

116964    40001688 

18.4932420 

6.9931906 

.002923977 

343 

117649  |   40353607 

18.5202592 

7.0000000 

.002915452 

344 

118336 

40707584 

18.5472370 

7.0067962 

.002906977 

345 

119025 

41063625 

18.57417'56 

7  0135791 

.002898551 

346 

119716 

41421736 

18.6010752 

7.0203490 

.002890173 

347 

120409 

41781923 

18.6279360 

7.0271058 

.002881844 

348 

121104 

42144192 

18.6547581 

7.0338497 

.002873563 

349 

121801 

42508549 

18.6815417 

7.0405806 

.002865330 

350 

122500 

42875000 

18.7082869 

7.04^2987 

.002857143 

351 

123201 

43243551 

18.7349940 

7.0540041 

.002849003 

352 

123904 

43614208 

18.7616630 

7.0600907 

.002840909 

353 

124609 

4398G977 

18.7882942 

7.0073707 

.002832861 

354 

125316 

44361864 

18.8148877 

7.0740440 

.002824859 

355 

126025 

44738875 

18.8414437 

7.0800988 

.002816901 

356 

126736 

45118016 

18.8679623 

7.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

7.0939709 

.002801120 

358 

128164 

45882712 

18.9208879 

7.1005885 

.002793296 

359 

128881 

46268279 

18.9472953 

7.1071937 

.002785515 

360 

129GOO 

46656000 

18.9736660 

7.1137866 

.002777778 

361 

130321 

47045881 

19,0000000 

7.1203074 

.002770083 

362 

131044 

47437928 

19.0262976 

7.1269360 

.002762431 

363 

131769 

47832147 

19.0525589 

7.1334925 

.002754821 

364 

132496 

48228544 

19.0787840 

7.140037'0 

.002747253 

365 

133225 

48627125 

19.1049732 

7.1465695 

.002739726 

366 

133956 

49027896 

19.1311265 

7.1530901 

.0027'32240 

367 

134689 

49430863 

19.1572441 

7.1595988 

.002724796 

368 

135424 

49836032 

19.1833261 

7.1660957 

.002717391 

369 

136161 

50243409 

19.2093727 

7.1725809 

.002710027 

370 

136900 

50653000 

19.2353841 

7.1790544 

.002702703 

371 

137641 

51064811 

19.2613603 

7.1855162 

.002695418 

372 

138384 

51478848 

19.2873015 

7.1919663 

.002688172 

136 


CUBE   ROOTS,   AND   RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

373 

139129 

51895117 

19.3132079 

7.1984050 

.002680965 

374 

139876 

52313624 

19.3390796 

7.2048322 

.002673797 

375 

140625 

52734375 

19.3649167 

7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194 

7.2176522 

.002659574 

377 

142129 

53582633 

19.4164878  1   7.2240450 

.002652520 

378 

142884 

54010152 

19.4422221  1   7.2304268 

.002645503 

379 

143641 

54439939 

19.4679223 

7.2367972 

.002638522 

380 

144400 

54872000 

19.4935887 

7.2431565 

.002631579 

381 

145161 

55306341 

19.5192213 

7.2495045 

.002624672 

382 

145924 

55742968 

19.5448203 

7.2558415 

.002617801 

383 

146689 

56181887 

19.5703858 

7.2621675 

.002610966 

884 

147'456 

56623104 

19.5959179 

7.2684824 

.002604167 

385 

148225 

57066625 

19.6214169 

7.2747864 

.002597403 

386 

148996 

57512456 

19.6468827 

7.2810794 

.002590674 

387 

149769 

57960603 

19.6723156 

7.2873617 

.002583979 

388 

150544 

58411072 

19.6977156 

7.2936330 

.002577320 

389 

151321 

58863869 

19.7230829 

7.2998936 

.002570694 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152881 

59776471 

19.7737199 

7.3123828 

.002557545 

392 

153664 

60236288 

19.7989899 

7.3186114 

.002551020 

393 

154449 

60698457 

19.8242276 

7.3248295 

.002544529 

394 

155236 

61162984 

19.8494332 

7.3310369 

.002538071 

395 

156025 

61629875 

19.8746069 

7.33?2339 

.002531646 

396 

156816 

62099136 

19.8997487 

7.3434205 

.002525253 

397 

157609 

62570773 

19.9248588 

7.3495966 

.002518892 

398 

158404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749844 

7.3619178 

.002506266 

400 

160000 

64000000 

20.0000000 

7.3680630 

.002500000 

401 

160801 

64481201 

20.0249844 

7.3741979 

.002493766 

402 

161604 

64964808 

20.0499377 

7.3803227 

.002487562 

403 

162409 

65450827 

20.0748599 

7.3864373 

.002481390 

404 

163216 

65939264 

20.0997512 

7.3925418 

.002475248 

405 

164025 

66430125 

20.1246118 

7.3986363 

.002469136 

406 

164836 

66923416 

20.1494417 

7.4047206 

.002463054 

407 

165649 

67419143 

20.1742410 

7.4107950 

.002457002 

408 

166464 

67917312 

20.1990099 

7.4168595 

.002450980 

409 

167281 

68417929 

20.2237484 

7.4229142 

.002444988 

410 

168100 

68921000 

20.2484567 

7.4289589 

.002439024 

411 

168921 

69426531 

20.2731349 

7.4349938 

.002433090 

412 

169744 

69934528 

20.2977831 

7.4410189 

.002427184 

413 

170569 

70444997 

20.3224014 

7.4470342 

.002421308 

414 

171396 

70957944 

20.3469899 

7.4530399 

.002415459 

415 

172225 

71473375 

20.3715488 

7.4590359 

.002409639 

416 

17'3056 

71991296 

20.3960781 

7.4650223 

.002403846 

417 

173889 

72511713 

20.4205779 

7.4709991 

.(Kt'398082 

418 

174724 

73034632 

20.4450483 

7.4769664 

.002392344 

419 

175561 

73560059 

20.4694895 

7.4829242 

.002386635 

420 

176400 

74088000 

20.4939015 

7.4888724 

.002380052 

421 

177241 

74618461 

80.5188845 

7.4948113 

.002375'J'ir 

422 

178084 

75151448 

80.6486888 

7.5007406 

iMi-j:«59668 

423 

178929 

75686967 

80.6669688 

7.5066607 

.002364066 

424 

179776 

AJ225024 

20  591  OT5 

7.5125715 

.002358491 

425 

180625 

76765625 

20.6155281 

7.5184730 

.002352!  HI 

426 

181476 

7730877'6 

20.6397674 

7.5243652 

.002347418 

427 

182329 

77854483 

20.6639783 

7.5808488 

.002341!  120 

428 

183184 

78402752 

80.6881609 

7.5861881 

86448 

429 

184041 

78953589 

80.7188169 

7.5410667 

.002331002 

430 

184900 

79507000 

20.7364414 

7.5478423 

85581 

431 

185761 

800(52991 

W).  7605395 

7.5586888 

,0088$ 

432 

186624 

80021568 

80.7846097 

7.5595863 

.01)2:11  IS  15 

433 

187489 

81188787 

•  80.  80W  r,;>() 

7.5658548 

09469 

434 

188356 

81746504 

80.8886667 

7.5711743 

.0'fJ304147 

137 


TABLE   X. — SQUARES,   CUBES,   SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
lioots. 

Cube  Roots. 

Reciprocals. 

435 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

i  190096 

82881856 

20.8806130 

7.5827865 

.002293578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.002288330 

438 

191844 

8402<67'2 

20.9284495 

7.5943633 

.002283105 

439 

192721 

84604519 

.20.9523268 

7.6001385 

.002277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.002272727 

441 

194481 

8576U121 

21.0000000 

7.6116626 

.002267574 

442 

195364 

86350888 

21.0237960 

7.6174116 

.002262443 

443 

196249 

86938307 

21.0475652 

7.6231519 

.002257'336 

444 

197136 

87528384 

21.0713075 

7.6288837 

.002252252 

445 

198025 

88121125 

21.0050231 

7.6346067 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.002242152 

447 

199809 

89314623 

21.1423745 

7.6460272 

.002237136 

448 

200704 

89915392 

21.1660105 

7.6517'247 

.002232143 

449 

201601 

90518849 

21.1896201 

7.6574133 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

203401 

91733851 

21.2367606 

7.6687665 

.002217295 

452 

204304 

923454J8 

21.2602916 

7.6744303 

.002212389 

453 

205209 

92959677 

SI.  2837967 

7.0800857 

.002207506 

454 

206116 

93576664 

21.3072758 

7.6857328 

.002202643 

455 

207025 

94196375 

21.3307290 

7.6913717 

.002197802 

456 

207936 

94818816 

21.3541565 

7.6970023 

.002192982 

457 

208849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

209764 

96071912 

21.4009346 

7.7082388 

.00218:3406 

459 

210681 

96702579 

21.4242853 

7.7138448 

.002178649 

460 

211600 

97336000 

21.4476106 

7.7194426 

.002173913 

461 

212521 

97972181 

21.4709106 

7.7250325 

.002169197 

462 

213444 

98611128 

21.4941853 

7.7306141 

.002164502  • 

463 

214369 

99252847 

21.5174348 

7.7361877 

.002159827 

464 

215296 

99897344 

21.5406592 

7.7417532 

.002155172 

465 

216225 

100544625 

21.5638587 

7.747'3109 

.002150538 

466 

217156 

101194696 

21.5870331 

7.7528606 

.002145923 

467 

218089 

101847563 

21.6101828 

7.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

7.7639361 

.002136752 

469 

219961 

103161709 

21.6564078 

7.7694620 

.002132196 

470 

220900 

103823000 

21.6794834 

7.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

7.7804904 

.002123142 

472 

222784 

105154048 

21.7255610 

7.7859928 

.002118644 

473 

223729 

105823817 

21.7485632 

7.7914875 

.002114165 

474 

224676 

106496424 

21.7715411 

7.7969745 

.002109705 

475 

225625 

107171875 

21.7944947 

7.8024538 

.002105263 

476 

226576 

107850176 

21.8174242 

7.8079254 

.002100840 

477 

227529 

108531333 

21.8403297 

7.8133892 

.002096436 

478 

228484 

109215352 

21  8032111 

7.8188456 

.002092050 

479 

229441 

109902239 

21.8860686 

7.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

7.8297353 

.002088333 

481 

231361 

111284641 

21.9317122 

7.8351688 

.002079002 

482 

232324 

111980168 

21.9544984 

7.8405949 

.002074689 

483 

233289 

112678587 

21.9772610 

7.8460134 

.002070393 

484 

234256 

113379904 

22.0000000 

7.8514244 

.002066116 

485 

235225 

114084125 

22.0227155 

7.8568281 

.002061856 

486 

236196 

114791256 

22.0454077 

7.8622242 

.002057613 

487 

237169 

115501303 

22.0680765 

7.8676130 

.002053388 

488 

238144 

116214272 

22.0907220 

7.8729944 

.002049180 

489 

239121 

116930169 

22.1133444 

7  8783684 

.002044990 

490 

240100 

117649000 

22.1&59436 

7.8837352 

.002040816 

491 

241081 

118370771 

22.1585198 

7.8890946 

.002036660 

492 

242064 

119095488 

22.1810730 

7.8944463 

.002032520 

493 

243049 

119823157 

22.2036033 

7.8997917 

.002028398 

494 

244036 

120553784 

22.2261108 

7.9051294 

.002024291 

495 

245025 

121287375 

22.2485955 

7.9104599 

.002020202 

496 

246016 

122023936 

22.2710575 

7.9157832 

.002016129 

138 


CUBE   ROOTS,   AtfD   RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

497 

247009 

122763473 

22.2934968 

7.9210994 

.002012072 

498 

248004 

123505992 

22.3159136 

7.9264085 

.002008032 

499 

249001 

124251499 

22.3383079 

7.9317104 

.002004008 

500 

250000 

125000000 

22.3606798 

7.9370053 

.002000000 

501 

251001 

125751501 

22.3830293 

7.9422931 

.001996008 

502 

252004 

126506008 

22.4053565 

7.9475739 

.001992032 

503 

253009 

127263527 

22.4276615 

7.9528477 

.001988072 

504 

254016 

128024064 

22.4499443 

7.9581144 

.001984127 

505 

255025 

128787625 

22.4722051 

7.9633743 

.001980198 

506 

256036 

129554216 

22.4944438 

7.9686271 

.001976285 

507 

257049 

13032:3843 

22.5166605 

7.9738731 

.001972387 

508 

258064 

131096512 

22.5388553 

7.9791122 

.001968504 

509 

259081 

131872229 

22.5610283 

7.9843444 

.001964637 

510 

260100 

132651000 

22.5831796 

7.9895697 

.001960784 

511 

261121 

133432831 

22.6053091 

7.9947883 

.001956947 

512 

262144 

134217728 

22.6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

514 

264196 

135796744 

22.6715681 

8.0104082 

.001945525 

515 

265225 

136590875 

22.6936114 

8.0155946 

.001941748 

516 

266256 

137388096 

22.7156334 

8.0207794 

.001937984 

517 

267289 

138188413 

22.7376340 

8.0259574 

.001934236 

518 

268324 

138991832 

22.7596134 

8.0311287 

.001930502 

519 

269361 

139798359 

22.7815715 

8.0362935 

.001926782 

520 

270400 

140608000 

22.  8035085 

8.0414515 

.001923077 

521 

271441 

141420761 

22.8254244 

8.0466030 

.0019193S6 

522 

272484 

142236648 

22.8473193 

8.0517479 

.001915709 

523 

273529 

143055667 

22.8691933 

8.0568862 

.001912046 

524 

274576 

143877824 

22.8910463 

8.0620180 

.001908397 

525 

275625 

144703125 

22.9128785 

8.0671432 

.001904762 

526 

276676 

145531576 

22.9346899 

8.0722620 

.001901141 

527 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

528 

278784 

147197952 

22.9782506 

8.0824800 

.001893939 

529 

279841 

148035889 

23.0000000 

8.0875794 

.001890359 

530 

280900 

148877000 

23.0217289 

8.0926723 

.001886792 

531 

281961 

149721291 

23.0434372 

8.0977589 

.001883239 

532 

283024 

150568768 

23.0651252 

8.1028390 

.001879699 

533 

284089 

151419437 

23.0867928 

8.1079128 

.001876173 

534 

285156 

152273304 

23.1084400 

8.1129803 

.001872(559 

535 

286225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

287296 

153990656 

23.1516738 

8.1230962 

.001805(572 

537 

288369 

154854153 

23.1732605 

8.1281447 

.001802197 

538 

289444 

155720872 

23.1948270 

8.1331870 

.001K58736 

539 

290521 

156590819 

23.2163735 

8.1382230 

.001855288 

540 

291600 

157464000 

23.2379001 

8.1432529 

.00185ia52 

541 

292681 

158340421 

23  .  2594067 

8.1488705 

.001848429 

542 

293764 

159220088 

23.2808935 

8.1532939 

.001845018 

543 

294849 

160103007 

23.3023604 

8.1583051 

.001841021 

544 

295936 

160989184 

23.3238076 

8.1033102 

.00183* 

545 

297025 

161878625 

23.  3452351 

S.10S3092 

.001834^2 

546 

298116 

162771336 

23.3666429 

8.1733020 

.00183151),! 

547 

299209 

163667323 

23.3880311 

8.173 

.001898151 

548 

300304 

1045005<>2 

23.40'.)399S 

8  1S32695 

.001924818 

549 

301401 

1U5469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

166375000 

23.4520788 

8.1932127 

.001S1R182 

551 

303601 

167284151 

23.4733892 

8.1981758 

.80181 

552 

304704 

16819000S 

23.49WSO:> 

8.2031319 

.IM1SH594 

553 

805809 

169112377 

23.5159520 

B.203 

.001808818 

554 

30001  6 

1700314(54 

83.5372046 

8.2130271 

.001805054 

555 

308025 

170953875 

23.5584380 

8.2179657 

.001801802 

556 

309136 

171879616 

23.579C.522 

8.2228985 

.00179* 

557 

310249 

172808(593 

23.  (5008474 

8.2-j; 

.00179 

558 

311364 

173741112 

23.6220236 

8.2327463 

.001792115 

139 


TABLE   X. — SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

559 

312481 

174676879 

23.6431808 

8.2376614 

001788909 

560 

313600 

175616000 

23.6643191 

8.2425706 

.001785714 

561 

314721 

176558481 

23.6854386 

8.2474740 

.001782531 

562 

315844 

177504328 

23.7065392 

8.2523715 

001779359 

563 

316969 

178453547 

23.7'276210 

8.2572633 

001776199 

564 

318096 

179406144 

23.7486842 

8.2621492 

.001773050 

565 

319225 

180362125 

23.7697286 

S.267'0294 

.001769912 

566 

320356 

181321496 

23.7907545 

8.2719039 

.001766784 

567 

321489 

182284263 

23.8117618 

8.2767726 

001763668 

568 

322624 

183250432 

23.8327506 

8.2816355 

,001760563 

569 

323761 

184220000 

&i.8537'209 

8.2864928 

.001757469 

570 

324900 

185193000 

23.8746728. 

8.2913444 

.001754386 

571 

326041 

186169411 

23.8956063 

8.2961903 

.001751313 

572 

327184 

187149248 

23.9165215 

8.3010304 

.001748252 

573 

328329 

188132517 

23.9374184 

8.3058651 

.001745201 

574 

329476 

189119224 

23.9582971 

8.3106941 

.001742160 

575 

330625 

190109375 

23.9791576 

8.3155175 

.001739130 

576 

331776 

191102976 

24.0000000 

8.3203353 

.001736111 

577 

332929 

192100033 

24.0208243 

8.3251475 

.001733102 

578 

334084 

193100552 

24.0416306 

8.3299542 

.001730104 

579 

335241 

194104539 

24.0624188 

.  8.3347553 

.001727116 

580 

336400 

195112000 

24.0831891 

8.3395509 

.001724138 

581 

337561 

196122941 

24.1039416 

8.3443410 

.001721170 

582 

338724 

197137368 

24.1246762 

8.3491256 

.001718213 

583 

339889 

198155287 

24.1453929 

8.3539047 

.001715266 

584 

341056 

199176704 

24.1660919 

8.3586784 

.001712329 

585 

342225 

200201625 

24.1867732 

8.3634466 

.001709402 

586 

343396 

201230056 

24.2074369 

8.3682095 

001706485 

587 

344569 

202262003 

24.2280829 

8.3729668 

.001703578 

588 

345744 

203297472 

24.2487113 

8.3777188 

.001700680 

589 

346921 

204336469 

24.2693222 

8.3824653 

.001697793 

590 

348100 

205379000 

24.2899156 

8.3872065 

.001694915 

591 

349281 

206425071 

24.3104916 

8.3919423 

.001692047 

592 

350464 

207474688 

24.3310501 

8.3966729 

.001689189 

593 

351649 

208527857 

24.3515913 

8.4013981 

.001686341 

594 

352836 

20S5S4584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

2127761  7'3 

24.4335834 

8.4202460 

.001675042 

598 

357604 

213847192 

24.4540385 

8.4249448 

.001672241 

599 

358801 

214921799 

24.4744765 

8.4296383 

.001669449 

600 

360000 

216000000 

24.4948974 

8.4348267 

.001666667 

601 

361201 

217081801 

24.5153013 

8.4390098 

.001663894 

602 

362404 

218167203 

24.5356883 

8.4436877 

.001661130 

603 

363609 

219256227 

24.5560583 

8.4483605 

.001658375 

604 

364816 

220348864 

24.5764115 

8.4530281 

.001055629 

605 

366025 

221445125 

24.5967478 

8.4576906 

.001652893 

606 

367236 

222545016 

24.6170673 

8.4623479 

.001650165 

607 

368449 

223648543 

24.6373,00 

8.4670001 

.001047446 

608 

369664 

224755712 

24.0576560 

8.4716471 

.001644737 

609 

3?'0881 

225866529 

24.6779254 

8.47'62b92 

.001642036 

610 

372100 

226981000 

24.6981781 

8.4809261 

.001639344 

611 

373321 

228099131 

24.7184142 

8.4855579 

.001030661 

612 

374544 

229220928 

24.7386338 

8.4901848 

001633987 

613 

375769 

230346397 

24.7588368 

8.4948065 

001631321 

614 

376996 

231475544 

24.7790234 

8.4994233 

.001028664 

615 

378225 

232608375 

24.7991935 

8.5040350 

.001626016 

616 

379456 

233744896 

24.8193473 

8.5086417 

.001623377 

617 

380689 

234885113 

24.8394847 

8.5132435 

.001620746 

618 

381924 

236029032 

24.8596058 

8.5178403 

.001618123 

619 

383161 

237176659 

24.8797106 

8.5224321 

.001615509 

620 

384400 

238328000 

24.8997992 

8.5270189 

.001612903 

140 


CUBE   BOOTS,   AKD   RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

621 
622 
623 
624 
625 
626 
627 
628 
629 

385641 
386884 
388129 
389376 
390625 
391876 
393129 
394384 
395641 

239483061 
240041848 
241804307 
242970024 
244140625 
245314376 
246491883 
247673152 
248858189 

24.9198710 
24.9399278 
24.9599079 
24.9799920 
25.0000000 
25.0199920 
25.0399081 
25.0599282 
25.0798724 

8.5316009 
8.5361780 
8.5407501 
8.5453173 
8.5498797 
8.5544372 
8.5589899 
8.5635377 
8!  5680807 

.001610300 
.001007717 
.001005130 
.001002564 
.001600000 
.001597444 
.001594896 
.001592357 
.001589825 

630 
631 
632 
633 
634 
635 
636 
637 
638 
639 

396900 
398101 
399424 
400089 
401956 
403225 
404496 
405709 
407044 
408321 

250047000 
251239591 
252435968 
253636137 
254840104 
256047875 
257259456 
258474853 
259694072 
260917119 

25.0998008 
25.1197134 
25.1390102 
25.1594913 
25.1793500 
25.1992003 
25.2190404 
25.2388589 
25.2580019 
25.2784493 

8.5726189 
8.5771523 
8.5816809 
8.5862047 
8.5907238 
8.5952380 
8.5997470 
8.6042525 
8.6087526 
8.6132480 

.001587302 
.001584786 
.001582278 
.001579779 
.001577287 
.001574803 
.001572327 
.001509859 
.001567398 
.001564945 

640 
641 
642 
643 
644 
645 
646 
647 
648 
649 

409000 
410881 
412164 
413449 
414736 
416025 
417316 
418609 
419904 
421201 

262144000 
263374721 
264609288 
265847707 
267089984 
268330125 
269586136 
270840023 
272097792 
273359449 

25.2982213 
25.3179778 
25.3377189 
25.3574447 
25.3771551 
25.3968502 
25.4165301 
25.4361947 
25.4558441 
25.4754784 

8.6177388 
8.6222248 
8.6267063 
8.6311830 
8.6356551 
8.6401220 
8.6445855 
8.6490437 
8.6534974 
8.6579465 

.001562500 
.001560062 
.001557632 
.001555210 
.001552795 
.001550388 
.001547988 
.001545595 
.001543210 
.001540832 

650 
651 
652 
653 
654 
655 
656 
657 
658 
659 

422500 
423801 
425104 
426409 
427716 
429025 
430336 
431649 
432964 
434281 

274625000 
.  275894451 
277167808 
278445077 
279726264 
281011375 
282300416 
283593393 
284890312 
286191179 

25.4950976 
25.5147016 
25.5342907 
25.5538647 
25.5734237 
25.5929678 
25.6124969 
25.6320112 
25.6515107 
25.6709953 

8.6623911 
8.6668310 
8.6712665 
8.6756974 
8.6801237 
8.6845456 
8.6889630 
8.6933759 
8.6977843 
8.7021882 

.001538462 
.001536098 
.001533742 
.001531394 
.001529052 
.001526718 
.001524390 
.001522070 
.001519757 
.001517451 

660 
661 
662 
663 
664 
665 
666 

435600 
436921 
438244 
439569 
440896 
442225 
443556 

287496000 
288804781 
290117528 
291434247 
292754944 
294079625 
295408296 

25.6904652 
25.7099203 
25.7293607 
25.7487864 
25.7681975 
25.7875939 
25.8069758 

8.7065877 
8.7109827 
8.7153734 
8.7197596 
8.7241414 
8.7285187 
8.7328918 

.001515152 
.001512a59 
.001510574 
.001508296 
.001506024 
.001503759 
.001501502 

667 
668 

444889 
446224 

296740963 
298077632 

25.8263431 
25.8456960 

8.7372604 
8.7416246 

.001499250 
.001497006 

669 

447561 

299418309 

25.8650343 

8.7459846 

.001494768 

670 
671 
672 
673 
674 
675 
670 

448900 
450241 
451584 
452929 
454276 
455625 
450976 

300763000 
302111711 
303464448 
304821217 
300182024 
307546875 
308915776 

25.8843582 
25.9036677 
25.9229028 
25.9422435 
25.9615100 
25.9807621 
26.0000000 

8.7503401 
8.7546913 
8.7590383 
8.7633809 
8.7677192 
8.7720532 
8.7763830 

.001492537 
.001490313 
.001488095 
.001485884 
.00148:3080 
.001481481 
.001479290 

677 

678 

458329 
459084 

310288733 
311665752 

26.0192237 
26.0384331 

8.7807084 
8.7850296 

.001477105 
.001474926 

679 

401041 

313046839 

26.0576284 

8.7893466 

.001472754 

680 

402400 

314432000 

26.0708096 

8.7936593 

.001470588 

681 

403701 

315821241 

26.0959767 

8.7979679 

.001408429 

682 

405124 

317214568 

26.1151297 

8.8022721 

.001400276 

141 


TABLE   X. — SQUARES,    CUBES,   SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

6&3 

466489 

318611987 

26.1342687 

8.8065722 

.001464129 

684 

467856 

320013504 

26.153-3937 

8.8108081 

.001401988 

685 

469225 

321419125 

26.1725047 

8.8151598 

.001459854 

686 

470596 

322828856 

26.1916017 

3.8194474 

.001457720 

687 

471969 

324242703 

26.2106848 

8.8237307 

.001455004 

688 

473344 

3251360672 

26.2297541 

8.8280099 

.001453488 

689 

474721 

3270827(59 

26.2488095 

8.8322850 

.001451379 

690 

476100 

328509000 

26.2678511 

8.8305559 

.001449275 

691 

477481 

329939371 

26.2868789 

8.8408227 

.001447178 

692 

478864 

331373888 

26.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8.8493440 

.001443001 

694 

481636 

334255384 

26.3438797 

8.8535985 

.001440922 

695 

483025 

335702375 

26.3628527 

8.8578489 

.001438849 

696 

484416 

337153536 

26.3818119 

8.8020952 

.001430782 

697 

485809 

338608873 

26.4007576 

8.8003375 

.001434720 

698 

487204 

340068392 

26.4196896 

8.8705757 

.001432065 

699 

488601 

341532099 

26.4386081 

8.8748099 

.001430015 

700 

490000 

343000000 

26.4575131 

8.8790400 

.001428571 

701 

491401 

344472101 

26.4764046 

8.8832001 

.001426534 

702 

492804 

345948408 

26.4952826 

8.8874882 

.001424501 

703 

494209 

347428927 

26.5141472 

8.8917063 

001422475 

704 

495616 

348913664 

26.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

26.5518361 

8.9001304 

.001418440 

706 

498436 

351895816 

26.5706605 

8.9043366 

.001416431 

707 

499849 

353393243 

26.5894716 

8.9085387 

.001414427 

708 

501264 

354894912 

20.6082094 

8.9127309 

.001412429 

709 

502081 

356400829 

20  6270539 

8.9109311 

.001410437 

710 

504100 

a57911000 

26.6458252 

8.9211214 

.001408451 

711 

505521 

359425431 

20.0045833 

8.9253078 

.001400470 

712 

506944 

360944128 

26.6833281 

8.9294902 

.001404494 

713 

508369 

362467097 

26.7020598 

8.9330087 

.001402525 

714 

509796 

363994344 

26.7207784 

8.9378433 

.001400500 

715 

511225 

365525875 

20.7394839 

8.9420140 

.001398001 

716 

512656 

367061696 

20.7581703 

8.9401809 

.001390648 

717 

514089 

368601813 

20.7768557 

8.9503438 

001394700 

718 

515524 

370146232 

26.7955220 

8.9545029 

.001392758 

719 

516961 

371694959 

26.8141754 

8.9580581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519841 

374805361 

20.8514432 

8.9009570 

001386963 

722 

521284 

376367048 

20.8700577 

8.9711007 

.001385042 

723 

522729 

377933067 

20.8886593 

8.9752400 

.001383126 

724 

524176 

379503424 

20.9072481 

8.9793706 

.001381215 

725 

525625 

381078125 

20.9258240 

8.9835089 

.001379310 

726 

527076 

382657176 

26.9443872 

8.9876373 

.001377410 

727 

528529 

384240583 

26.9029375 

8.9917020 

.001375516 

728 

529984 

385828352 

20.9814751 

8.9958829 

.001373026 

729 

531441 

387420489 

27.0000000 

9.0000000 

.001371742 

730 

532900 

389017000 

27.0185122 

9.0041134 

.001369803 

731 

534361 

390617'891 

27.0370117 

9.0082229 

.001307989 

732 

535824 

392223168 

27.0554985 

9.0123288 

.001300120 

733 

537289 

393832837 

27.0739727 

9.0104309 

.001304256 

734 

538756 

395446904 

27.0924344 

9.0205293 

.001302398 

735 

540225 

397065375 

27.1108834 

9.0240239 

.001300544 

736 

541696 

398688256 

27.1293199 

9.0287149 

.001358090 

737 

543169 

400315553 

27.1477439 

9.0328021 

.001350852 

738 

544644 

401947272 

27.1001554 

9.0308857 

.001355014 

739 

546121 

403583419 

27.1845544 

9.0409055 

.001353180 

740 

547COO 

405224000 

27.2029410 

9.0450419 

.001351351 

741 

549081 

406869021 

27.2213152 

9.0491142 

.001349528 

742 

550564 

408518488 

27.2390709 

9.0531831 

.001347709 

.  743 

552049 

410172407 

27.2580203 

9.0572482 

.001345895 

744 

553536 

411830784 

27.2703034 

9.0013098 

.001344086 

\42 


CUBE   ROOTS,    AND    RECIPROCALS. 


fNo. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

745 

555025 

413493625 

27.2946881 

9.0653677 

.001342282 

746 

556516 

415160936 

27.3130006 

9.0694220 

.001340483 

747 

558009 

416832723 

27.3313007 

9.0734726 

.001338688 

748 

559504 

418508992 

27.3495887 

9.0775197 

.001336898 

749 

561001 

420189749 

27.3678644 

9.0815631 

.001335113 

750 

562500 

421875000 

27.3861279 

9.0856030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.0890392 

.Oi'  1331558 

752 

565504 

425259008 

27.4226184 

9.0936719 

.001329787 

753 

567009 

426957777 

27.4408455 

9.0977010 

.001328021 

754 

568516 

428661064 

27.4590604 

9.1017205 

.001326260 

755 

570025 

430368875 

27.4772633 

9.1057485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1097669 

.001322751 

757 

573049 

433798093 

27.5136330 

9.1137818 

.001321004 

758 

574564 

435519512 

27.5317998 

9.1177931 

.001319261 

759 

576081 

437245479 

27.5499546 

9.1218010 

.001317523 

760 

577600 

438976000 

27.5680975 

9.1258053 

.001315789 

761 

579121 

440711081 

27.5862284 

9.1298061 

.001314060 

762 

580644 

442450728 

27.6043475 

'  9.1338034 

.001312336 

763 

582169 

444194947 

27.62,4546 

9.1377971 

.001310616 

764 

583696 

445943744 

27.6405499 

9.1417874 

.001308901 

765 

585225 

447697125 

27.6586334 

9.1457742 

.001307190 

766 

586756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

588289 

451217663 

27.6947648 

9.1537375 

.001303781 

768 

589824 

452984832 

27.7128129 

9.1577139 

.001302083 

769 

591361 

454756609 

27.7308492 

9.1616869 

.001300390 

770 

592900 

456533000 

27.7488739 

9.1656565 

.001298701 

771 

594441 

458314011 

27.76688C8 

9.1696225 

.001297017 

772 

595984 

460099648 

27.7848880 

9.1735852 

.001295337 

773 

597529 

461889917 

27.8028775 

9.1775445 

.001293661 

774 

599076 

463684824 

27.8208555 

9.1815003 

.001291990 

775 

600625 

465484375 

27.8388218 

9.1854527 

.001290323 

776 

602176 

467288576 

27.8567766 

9.1894018 

.001288660 

777 

603729 

469097433 

27.8747197 

9.1933474 

.001287001 

778 

605284 

470910952 

27.8926514 

9.1972897 

.001285347 

779 

606841 

472729139 

27.9105715 

9.2012286 

.001283697 

780 

608400 

474552000 

27.9284801 

9.2051641 

.001282051 

781 

609961 

476379541 

27.9463772 

9.2090902 

.001280410 

782 

611524 

478211768 

27.9642629 

9.2130250 

.001278772 

783 

613089 

480048687 

27.982137'2 

9.2169505 

.001277139 

784 

614656 

481890304 

28.0000000 

9.2208726 

.001275510 

785 

616225 

483736625 

28.0178515 

9.2247914 

.001273885 

786 

617796 

485587656 

28.0356915 

9.2287068 

.00127'2265 

787 

619369 

4874434CS 

28.0535203 

9.2326189 

.00127'0648 

788 

620944 

480906879 

28.0713377 

9.2365277 

.001269036 

789 

622521 

491169069 

28.0891438 

9.2404333 

.001267427 

790 

624100 

493039000 

28.1069386 

9.2443355 

.001265823 

791 

625681 

494913671 

28.1247222 

9.2482344 

.001264223 

792 

627264 

496793088 

28.1424946 

9.25213DO 

.001262626 

793 

628849 

498677257 

28.1602557 

9.2560224 

.001261034 

794 

630436 

500566184 

28.1780056 

9.2599114 

.001259446 

795 

632025 

502459875 

28.1957444 

9.2637973 

.001257862 

796 

633616 

504358336 

28.2134720 

9.2676798 

.001256281 

797 

635209 

506261573 

28.2311884 

9.2715592 

.001254705 

798 

636804 

508169592 

28.2488938 

9.2754352 

.001253133 

799 

638401 

510082399 

28.2665881 

9.2793081 

.001251564 

800 

640000 

512000000 

28.2842712 

9.2831777 

.001250000 

801 

641601 

513922401 

28.3019434 

9.2870440 

.001248439 

802 

643204 

515849(508 

28.3196045 

9.2909072 

.001246883 

803 

644809 

517781627 

28.3372546 

9.2947671 

.001245330 

804 

046416 

519718464  1  28.3548938 

9.2986239 

.001243781 

805 

648025 

521660125  i  28.3725219 

9.3024775 

.001242236 

806 

649636 

523606616    28.3901391 

9.3063278 

.001240695 

143 


TABLE   X. — SQUARES,    CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

807 

651249 

525557943 

28.4077454 

9.3101750 

.001239157 

808 

652864 

527514112 

28.4253408 

9.3140190 

.001237624 

809 

654481* 

529475129 

28.4429253 

9.3178599 

.001236094 

810 

656100 

531441000 

28.4604989 

9.3216975 

.001234568 

811 

657721 

533411731 

28.4780617 

9.3255320 

.001233046 

812 

659344 

535387328 

28.4956137 

9.3293634 

.001231527 

813 

660969 

537367797 

28.5131549 

9.3331916 

.001230012 

814 

662596 

539353144 

28.5306852 

9.3370167 

.001228501 

815 

664225 

541343375 

28.5482048 

9.3408386 

.001226994 

816 

665856 

543338496 

28.5657137 

9.3446575 

.001225490 

817 

667489 

545338513 

28.5832119 

9.3484731 

.001223990 

818 

669124 

547343432 

28.6006993 

9.3522857 

.001222494 

819 

670761 

549353259 

28.6181760 

9.3560952 

.001221001 

820 

672400 

551368000 

28.6356421 

9.3599016 

.001219512 

821 

674041 

553387661 

28.6530976 

9.3637049 

.001218027 

822 

675684 

555412248 

28.6705424 

9.3675051 

.001*216545 

823 

677329 

557441767 

28.6879766 

9.3713022 

.001215067 

824 

678976 

559476224 

28.7054002 

9.3750963 

.001213592 

825 

680625 

561515625 

28.7228132 

9.3788873 

.001212121 

826 

682276 

563559976 

28.7402157 

9.3826752 

.001210654 

827 

683929 

565609283 

28.7576077 

9.3864600 

.001209190 

828 

685584 

567663552 

28.7749891 

9.3902419 

.001207729 

829 

687241 

569722789 

28.7923601 

9.3940206 

.001206273 

830 

688900 

571787000 

28.8097206 

9.3977964 

.001204819 

831 

690561 

573856191 

28.8270706 

9.4015691 

.001203369 

832 

692224 

575930368 

28.8444102 

9.4053387 

.001201923 

833 

693889 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

28.8790582 

9.4128690 

.001199041 

835 

697225 

582182875 

28.8963666 

9.4166297 

.001197605 

836 

698896 

584277056 

28.9136646 

9.4203873 

.001196172 

837 

700569 

586376253 

28.9309523 

9.4241420 

.001194743 

838 

702244 

588480472 

28.9482297 

9.4278936 

.001193317 

839 

703921 

590589719 

28.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

28.9827535 

9.4353880 

.001190476 

841 

707281 

594823321 

29.0000000 

9.4391307 

.001189061 

842 

708964 

596947688 

29.0172363 

9.4428704 

.001187648 

843 

710649 

599077107 

29.0344623 

9.4466072 

.001186240 

844 

712336 

601211584 

29.0516781 

9.4503410 

.001184834 

845 

714025 

603351125 

29.0688837 

9.4540719 

.001183432 

846 

715716 

605495736 

29.0860791 

9.4577999 

.001182033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001180638 

848 

719104 

609800192 

29.1204396 

9.4652470 

.001179245 

849 

720801 

611960049 

29.1376046 

9.4689661 

.001177856 

850 

722500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

9.4763957 

.001175088 

852 

725904 

618470208 

29.1890390 

9.4801061 

.001173709 

853 

727609 

620650477 

29.2061637 

9.4838136 

.001172333 

854 

729316 

622835864 

29.2232784 

9.4875182 

.001170960 

855 

731025 

625026375 

29.2403830 

9.4912200 

.001169591 

856 

732736 

627222016 

29.2574777 

9.4949188 

.001168224 

857 

734449 

629422793 

29.2745623 

9.4986147 

.001166861 

858 

736164 

631628712 

29.2916370 

9.5023078 

.001165501 

859 

737881 

633839779 

29.3087018 

9.5059980 

.001164144 

860 

739600 

636056000 

29.3257566 

9.5096854 

.001162791 

861 

741321 

638277381 

29.3428015 

9.5133699 

.001161440 

862 

743044 

640503928 

29.3598365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3768616 

9.5207303 

.001158749 

864 

746496 

644972544 

29.3938769 

9.5244063 

.001157407 

865 

748225 

647214625 

29.4108823 

9.5280794 

.001156069 

866 

749956 

649461896 

29.4278779 

9.5317497 

.001154734 

867 

751689 

651714363 

29.4448637 

9.5354172 

.001153403 

868 
r 

753424 

653972032 

29.4618397 

9.5390818 

,001152074 

144 


CUBE    ROOTS,   AND    RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

8G9 

755161 

656234909 

29.4788059 

9.5427437    .001150748 

870 

756900 

658503000 

29.4957624 

9.5464027 

.001149425 

871 

758641 

660776311 

29.5127091 

9.5500589 

.001148106 

872 

760384 

663054848 

29.5296461 

9.5537123 

.001146789 

873 

762129 

665338617 

29.5465734 

9.5573630 

.001145475 

874 

763876 

667627624 

29.5634910 

9.5610108 

.001144165 

875 

765625 

669921875 

29.5803989 

9.5646559 

.001142857 

876 

767376 

672221376 

29.5972972 

9.5682982 

.001141553 

877 

769129 

674526133 

29.6141858 

9.5719377 

.001140251 

878 

770884 

676836152 

29.6310648 

9.5755745 

.001138952 

879  ' 

772641 

679151439 

29.6479342 

9.5792085 

.001137656 

880 

774400 

681472000 

29.6647939 

9.5828397 

.001136364 

881 

776161 

683797841 

29.6816442 

9.5864682 

.001135074 

882 

777924 

686128968 

29.6984848 

9.5900939 

.001133787 

883 

779689 

688465387 

29.7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973373 

.001131222 

885 

783225 

693154125 

29.7489496 

9.6009548 

.001129944 

886 

784996 

695506456 

29.7657'521 

9.6045696 

.001128668 

887 

786769 

697864103 

29.7825452 

9.6081817 

.001127396 

888 

788544 

700227072 

29.7993289 

9.6117911 

.001126126 

889 

790321 

702595369 

29.8161030 

9.6153977 

.001124859 

890 

792100 

704969000 

29.8328678 

9.6190017 

.001123596 

891 

793881 

707347971 

29.8496231 

9.6226030 

.001122334 

892 

795664 

7097'32288 

29.8663690 

9.6262016 

.001121076 

893 

797449 

712121957 

29.8831056 

9.6297975 

.001119821 

894 

799236 

714516984 

29.8998328 

9.6333907 

.001118568 

895 

801025 

716917375 

29.9165506 

9.6369812 

.001117318 

896 

802816 

719323136 

29.9332591 

9.6405690 

.001116071 

897 

804609 

721734273 

29.9499583 

9.6441542 

.001114827 

898 

806404 

724150792 

29.9666481 

9.6477367 

.001113586 

899 

808201 

726572699 

29.9833287 

9.6513166 

.001112347 

900 

810000 

729000000 

30.0000000 

9.6548938 

.001111111 

901 

811801 

731432701 

30.0166620 

9.6584684 

.001109878 

902 

813604 

7'33870808 

30.0333148 

9.6620403 

.001108647 

903 

815409 

736314327 

30.0499584 

9.6656096 

.001107420 

904 

817216 

738763264 

30.0665928 

9.6691762 

.001106195 

905 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001103753 

907 

822649 

746142643 

30.1164407 

9.6798604 

.001102536 

908 

824464 

748613312 

30.1330383 

9.6&34166 

.001101322 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1662063 

9.6905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.6940694 

.001097695 

912 

831744 

758550528 

30.1993377 

9.6976151 

.001096491 

913 

833569 

761048497 

30.2158899 

9.7011583 

.001095290 

914 

835396 

763551944 

30.2324329 

9.7046989 

.001094092 

915 

837225 

766060875 

30.2489669 

9.7082369 

.001092896 

916 

'  839056 

768575296 

30.2654919 

9.7117723 

.001091703 

917 

840889 

771095213 

30.2820079 

9.7153051 

.001090513 

918 

842724 

773620632 

30.2985148 

9.7188354 

.001089325 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

920 

846400 

778688000 

30.3315018 

9.7258883 

.001086957 

921 

848241 

781229961 

30.3479818 

9.7294109 

.001085776 

922   ;   850084 

783777448 

30.3644529 

9.7329309 

.001084599 

923 

851929 

786330467 

30.3809151 

9.7364484 

.001083423 

924 

853776 

788889024 

30.3973683 

9.7899684 

.001082251 

925 

855625 

791453125 

30.4138127 

9.7434758 

.001081081 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 

859329 

796597983 

30.4466747 

9.7504930 

.001078749 

928 

861184 

79!  >1  78752 

30.4630924 

9.7539979 

.001077586 

929 

863041 

801765089 

30.4795013  i   9.7575002 

.001076426 

930 

864900 

804357000 

30.4959014 

9.7610001    .001075269 

145 


TABLE   X. — SQUARES,   CUBES,    SQUARE   ROOTS, 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

931 

866761 

806954491 

30.5122926 

9.7644974 

.001074114 

932 

868624 

809557568 

30.5286750 

9.7679922 

.00107'2961 

933 

870489 

812166237 

30.5450487 

9.7714845 

.001071811 

934 

872356 

814780504 

30.5614136 

9.77'49743 

.001070664 

935 

874225 

817400375 

30.5777697 

9.7784616 

.001069519 

936 

876096 

820025856 

30.5941171 

9.7819466 

.001068376 

937 

877969 

822656953 

30.6104557 

9.7854288 

.001067236 

938 

879844 

825293672 

30.6267857 

9.7889087 

.00100tt)98 

839 

881721 

827936019 

30.6431069 

9.7923861 

.001064963 

940 

883600 

830584000 

30.6594194 

9.7958611 

.001063830 

941 

885481 

833237621 

30.6757233 

9.7993336 

.001062699 

942 

887364 

835896888 

30.6920185 

9.8028036 

.001061571 

943 

889249 

838561807 

80.7083051 

9.8062711 

.001060445 

944 

891136 

841232384 

cO.  7245830 

9.8097'362 

.001059322 

945 

893025 

843908625 

30.7408523 

9.8131989 

.001058201 

946 

894916 

846590536 

30.7571130 

9.8166591 

.001057082 

947 

896809 

849278123 

30.7733651 

9.8201169 

.001055966 

948 

898704 

851971392 

30.7896086 

9.8235723 

.001054852 

949 

900601 

854670349 

30.8058436 

9.8270252 

.001053741 

950 

902500 

857375000 

30.8220700 

9.8304757 

.001052632 

951 

904401 

860085351 

80.8382879 

9.8339238 

.001051525 

952 

906304 

862801408 

80.8544972 

9.8373695 

.001050420 

953 

908209 

865523177 

30.8706981 

9.8408127 

.001049318 

954 

910116 

868250664 

30.8868904 

9.8442536 

.001048218 

955 

912025 

87098387'5 

30.9030743 

9.8476920 

.001047120 

956 

913936 

873722816 

80.9192497 

9.8511280 

.001040025 

957 

915849 

876467493 

30.9354166 

9.8545617 

.001044932 

958 

917764 

879217912 

30.9515751 

9.8579929 

.001043841 

959 

919681 

881974079 

30.9677251 

9.8614218 

.001042753 

960 

921600 

884736000 

30.9838668 

9.8648483 

.001041667  . 

961 

923521 

887503681 

31.0000000 

9.8682724 

.001040583 

962 

925444 

890277128 

31.0161248 

9.8716941 

.001039501 

963 

927369 

893056347 

81.0322413 

9.8751135 

.001038422 

9&4 

929296 

895841344 

31.0483494 

9.87'85305 

.001037344 

965 

931225 

896682125 

31.0644491 

9.8819451 

.001036269 

966 

933156 

901428000 

31.0805405 

9.8853574 

.001035197 

967 

935089 

904231063 

31.0966236 

9.8887673 

.001034126 

968 

937024 

907039232 

81.1126984 

9.8921749 

.001033058 

969 

938961 

909853209 

31.1287648 

9.8955801 

.001031992 

970 

940900 

912673000 

31.1448230 

6.  8989830 

.001030928 

971 

942841 

915498611 

31.1608729 

9.9028835 

.001029866 

972 

944784 

918330048 

31.1769145 

9.9057817 

.001028807 

973 

946729 

921167317 

31.1929479 

9.9091776 

.001027749 

974 

948676 

924010424 

31.2089731 

9.9125712 

.001026694 

975 

950625 

926859375 

31.2249900 

9.9159624 

.001025641 

976 

952576 

929714176 

31.2409987 

9.9193513 

.001024590 

977 

954529 

932574833 

31.2569992 

9.9227379 

.001023541 

978 

956484 

935441352 

31.2729915 

9.9261222 

.001022495 

979 

958441 

938313739 

31.2889757 

9.9295042 

.001021450 

980 

960400 

941192000 

31.3049517 

9.9328839 

.001020408 

981 

962361 

944076141 

31.3209195 

9.9362613 

.001019368 

982 

964324 

946966168 

31.3368792 

9.9396363 

.001018330 

983 

966289 

949862087 

31.3528308 

9.9430092 

.001017294 

984 

968256 

952763904 

31.3687743 

9.9463797 

.001016260 

985 

970225 

956671625 

31.3847097 

9.9497479 

.001015228 

986 

972196 

958585256 

31.4006369 

9.9531138 

.001014199 

987 

974169 

901504803    31.4165561 

9.9564775 

.001013171 

988 

976144 

964430272  :  31.4324673 

9.9598389 

.001012146 

989 

978121 

967361669    31.4483704 

9.9631981 

.001011122 

990 

980100 

97'OS99000    31.4642654 

9.9665549 

.001010101 

991 

982081 

073242271    31.4801525 

9.9699095 

.001009082 

992 

984064 

9?'6191488    31.4960315 

9.9732619 

.001008065 

146 


CUBE  ROOTS,  AtfD  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

993 

986049 

979146657 

31.5119025 

9.9766120 

.001007049 

994 

988036 

982107784 

31.5277655 

9.9799599 

.001006036 

995 

990025 

985074875 

31.5436206 

9.9833055 

.001005025 

996 

992016 

988047936 

31.5594677 

9.9866488 

.001004016 

997 

994009 

991036973. 

31.5753068 

9.9899900 

.001003009 

998 

996004 

994011992 

31.5911380 

9.9933289 

.001002004 

999 

998001 

9970029SJ9  !  31  .6069613 

9.9966656 

.001001001 

1000 

1000000 

1000000000 

31.6227766 

10.0000000 

.001000000 

1001 

1002001 

1003003001 

31.6385840 

10.00&3322 

.0009990010 

1002 

1004004 

1006012008 

31.6543836 

10.0066622 

.0009980040 

1003 

1006009 

'  1009027027 

31.6701752 

10.0099899 

.0009970090 

loot 

1008016 

1012J48064 

31.6859590 

10.0133155 

.0009960159 

1003 

1010025 

1015075125    31.7017349 

10.0166389 

.0009950249 

1006 

1012036 

1018108216    31.7175030 

10.0199601 

.0009940358 

ioor 

1014049 

1021147343    31.7332633 

10.0232791 

.0009930487 

1003 

1016064 

1024192512  j  31.7490157 

10.0265958 

.0009920635 

1009 

1018081 

1027243729    31.7647603 

10.0299104 

.0009910803 

1010 

1020100 

1030301COJ  i  31.7804972 

10.0332228 

.0009900990 

1011 

1022121 

1033364331    31.7962262 

10.0365330 

.0009891197 

1012 

1024144 

1036433728 

31.8119474 

10.0398410 

.0009881423 

1013 

1026169 

1039509197 

31.8276609 

10.0431469 

.0009871668 

1014 

1028196 

1042590744 

31.8433666 

10.0464506 

.0009861933 

1015 

1030225 

1045678375 

31.8590646 

10.0497521 

.0009852217 

1016 

1032256 

1048772096 

31.8747549 

10.0530514 

.0009842520 

1017 

1034289 

1051871913 

31.8904374 

10.0563485 

.0009832842 

1018 

1036324 

1054977832 

31.9061123 

10.0596435 

.0009823183 

1019 

1038361 

1058089859 

31.9217794 

10.0629364 

.0009813543 

1020 

1040400 

1061208000 

31.9374388 

10.0662271 

.0009803922 

1021 

1042441 

1064332261 

31.9530906 

10.0695156 

.0009794319 

1022 

1044484 

1067462648 

31.9637317 

10.0728020 

.0009784736 

1023 

1046529 

1070599167 

31.9843712 

10.0760863 

.0009775171 

1021 

1048576 

1073741824 

32.1000000 

10.0793684 

.0009765625 

1025 

1050625 

1076890625 

32.0156212 

10.0826484 

.0009756098 

10215 

1052676 

1080045576 

32.0312348 

10.0859262 

.0009746589 

1027 

1054729 

1083206683 

32.0468407 

10.0892019 

.0009737098 

1028 

1056784 

1036373952 

32.0624391 

10.0924755 

.0009727626 

1029 

1058841 

1039547339 

32.0780298 

10.0957469 

.0009718173 

1030 

1060900 

1092727000 

32.0936131 

10.0990163 

.0009708738 

1031 

1062961 

1095912791 

32.1091887 

10.1022835 

.0009699321 

1032 

1065024 

1099104768 

32.1247568 

10.1055487 

.00  9689922 

1033 

1067089 

1102302937 

32.1403173 

10.1088117 

.0009680542 

1034 

1069156 

1105507304 

32.1558704 

10.1120726 

.0009671180 

1035 

1071225 

1103717875 

32.1714159 

10.1153314 

.0009661836 

1036 

1073296 

1111934656 

32.1869539 

10.1185882 

.0009652510 

1037 

1075369 

1115157653 

32.2024844 

10.1218428 

.0009643202 

1038 

1077444 

1118386872 

32.2180074 

10.1250953 

.000963:3911 

1039 

1079521 

1121622319 

32.2335229 

10.1283457 

.0009624639 

1040 

1081600 

1124864000 

32.2490310 

10.1315941 

.0009615385 

1041 

10a3681 

1128111921 

32.2645316 

10.1348403 

.0009606148 

1042 

1085764 

1131366088 

32.2800248 

10.1380845 

.0009596929 

1043 

1087849 

1134626507 

32.2955105 

10.1413266 

.0009587738 

1044 

1089936 

1137893184 

32.3109888 

10.1445667 

.0009578544 

1045 

1092025 

1141166125 

32.3264598 

10.1478047 

.0009569378 

1046 

1094116 

1144445336 

32.3419233 

10.1510406 

.0009560229 

1047 

1096209 

1147730823 

32.3573794 

10.1542744 

.0009551098 

1048 

1098304 

1151022592 

32.3728981 

10.1575062 

.0009541985 

1049 

1100401 

1154320649 

32.3882<>95 

10.1607359 

.0009532888 

1050 

1102500 

1157625000 

32.4037035 

10.1639636 

.0009523810 

1051 

1104601 

1160966651 

32.4191301 

10.1671893 

.0009514748 

1052 

1106704 

1164252608 

32.4345495 

10.1704129 

.0009505703 

1053 

1108809 

1167575877 

32.4499615 

10.1736344 

.0009496676 

1054 

1110916 

1170905464 

32.4653662 

10.1768539 

.0009487666 

147 


TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  100  L.  000.] 

_Xo.  109  L.  040. 

N. 

0 

1 

284 

5 

6 

7 

8 

9 

Diff. 

100 

000000 

0434 

0868     1301      1734 

2166 

2598 

3029 

3461 

3891 

432 

1 

4321 

4751 

5181      5609     6038 

6466 

6894 

7321 

7748 

8174 

428 

2 

8600 

9026 

9151      9876 

0300 

0724 

1147 

1570 

1993 

2415 

AOA 

3 

012a37 

3259 

3680  !  4100     4521 

4940 

5360 

5779 

6197 

6616 

420 

4 

4 

7033 

7451 

7868     8284     8700 

9116 

9532 

9947 

0361 

0775 

416 

i     5 

021189 

1603 

2016     2428     2841 

3252 

3664 

4075 

4486 

4896 

412 

6 

5306 

5715 

6125     6533     6942 

7350 

7757 

8164 

8571 

8978 

408 

7 

9384 

9789 

1 

0195     0600     1004 

1408 

1812 

2216 

261S 

3021 

404 

8   ' 

033424 

3826 

4227     4628     5029 

5430 

5830 

6230 

6629 

7028 

400 

0 

7426 

7825 

8223     8620     9017 

9414 

9811 

04 

0207 

0602 

0998 

397 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

434 

43.4 

86.8 

130.2 

173.6 

217.0 

260.4 

3C 

)3.8 

347.2 

390.6 

433 

43.3 

86.6 

129.9 

173.2 

216.5 

259.8 

31 

18.1 

346.4 

389.7 

432 

43.2 

86.4 

129.6 

172.8 

216.0 

259 

2 

ac 

>2.4 

345.6 

388.8 

431 

43.1 

86.2 

129.3 

172.4 

215.5 

258 

6 

301.7 

344.8 

387.9 

430 

i  43.0 

86.0 

129.0 

172.0 

215.0 

258 

0 

3( 

)1.0 

344.0 

387.0 

429 

42.9 

85.8 

128.7 

171.6 

214.5 

257.4 

300.3 

343.2 

386.1 

428 

42.8 

85.6 

128.4 

171.2 

214.0 

256 

8 

299.6 

342.4 

385.2 

427 

42.7 

85.4 

128.1 

170.8 

213.5 

256 

2 

2J 

)8.9 

341.6 

384.3 

426 

i  42.6 

85.2 

127.8 

170.4 

213.0 

255 

6 

298.2 

340.8 

383.4 

425 

42.5 

85.0 

127.5 

170.0 

212.5 

255.0 

297.5 

340.0 

382.5 

424 

42.4 

84.8 

127.2 

169.6 

212.0 

254 

4 

296.8 

339.2 

381.6 

423 

42.3 

84.6 

126.9 

169.2 

211.5 

253 

8 

21 

\Q   1 

338.4 

380.7 

422 

i  42.2 

84.4 

126.6 

168.8 

211.0 

253 

2 

2< 

»*.4 

337.6 

379.8 

421 

!  42.1 

84.2 

126.3 

168.4 

210.5 

252.6 

294.7 

336.8 

378.9 

420 

42.0 

84.0 

126.0 

168.0 

210.0 

252.0 

294.0 

336.0 

378.0 

419 

i  41.9 

83.8 

125.7 

167.6 

209.5 

251 

4 

2( 

J3.3 

335.2 

377.1 

418 

41.8 

83.6 

125.4 

167.2 

209.0 

250.8 

292.6 

334.4 

376.2 

417 

41.7 

as.  4 

125.1 

166.8 

208.5 

250 

2 

2 

)1.9 

333.6 

375.3 

416 

41.6 

83.2 

124.8 

166.4 

208.0 

249 

6 

291.2 

332.8 

374.4 

415 

41.5 

as.o 

124.5 

166.0 

207.5 

249 

0 

290.5 

332.0 

373.5 

414 

41.4 

82.8 

124.2 

165.6 

207.0 

248 

4 

289.8* 

331.2 

372.6 

413 

41.3 

82.6 

123.9 

165.2 

206.5 

247 

8 

2 

39.1 

330.4 

371.7 

412 

41.2 

82.4 

123.6 

164.8 

206.0 

247 

2 

2 

38.4 

329.6 

370.8 

411 

41.1 

82.2 

123.3 

164.4 

205.5 

246 

6 

21 

37.7 

328.8 

369.9 

410 

41.0 

82.0 

123.0 

164.0 

205.0       246 

0 

287.0 

328.0 

369.0 

409 

40.9 

81.8 

122.7 

163.6 

204.5 

245 

.4 

a 

36.3 

327.2 

368.1 

408 

40  8 

81.6 

122.4 

163.2 

204.0 

244 

.8 

285.6 

326.4 

367.2 

407 

40.7 

81.4 

122.1 

162.8 

203.5 

244 

.2 

284.9 

325.6     366.3 

406 

!  40.6       81.2 

121.8 

162.4 

203.0 

243 

6 

21 

34.2 

324.8  i  365.4 

405 

40.5       81.0 

121.5 

162.0 

202.5 

243.0 

283.5 

324.0  ,  364.5 

404 

!  4Q.4 

80  8 

121.2 

161.6 

202.0 

242 

.4 

282.8 

323.2     363.6 

403 

!  40.3       80.6 

120.9 

161.2 

201.5 

241 

.8 

282-.  1 

322.4     362.7 

402 

!  40.2       80.4 

120.6 

160.8 

201.0 

241 

2 

21 

31.4 

321.6     361.8 

401 

40.1 

80.  g 

120.3 

160.4 

200.5 

240 

.6 

21 

30.7. 

320.8     360.9 

400 

40.0 

80-0 

120.0 

160.0 

200.0 

240.0 

280.0 

320.0     360.0 

399 

39.9 

79.  £ 

1 

119.7 

159.6 

199.5 

239 

.4 

21 

j*9  3 

319.2     359.1 

398 

39.8       79.6 

119.4 

159.2 

199.0 

238 

.8 

278.6 

318.4     358.2 

397 

39.7       79.4 

[ 

119.1 

158.8 

198.5 

238 

.2 

o 

r7.9 

317.6     357.3 

396 

39.6 

79.  i 

i 

118.8 

158.4 

198.0 

237 

.6 

2 

rr.2 

316.8  1  356.4 

395 

39.5 

79.0         118.5 

158.0 

197.5       237 

.0 

276.5       316.0  1  355.5 

149 


TABLE    XI. — LOGARITHMS    OF   NUMBERS. 


No. 

110  L.  041.] 

[No.  119  L.  078. 

N. 

0 

1 

2 

3          4 

5 

6 

7 

8 

9 

Diff. 

110 

041393 

1787 

2182 

2576     2969 

3362 

3755 

4148 

4540 

4932 

393 

1 

5323 

5714 

6105 

6495     6885 

7275 

7664 

8053 

8442 

8830 

390 

2 

9218 

9606 

9993 

0380     0766 

1153 

1538 

1924 

2309 

2694 

386 

3 

053078 

3463 

3846 

4230     4613 

4996 

5378 

5760 

6142 

6524 

383 

4 

6905 

7286 

7666 

8046     8426 

8805 

9185 

9563 

9942 

0320 

379 

5 

060698 

1075 

1452 

1829     2206 

2582 

2958 

3333 

37'09 

4083 

376 

6 

4458 

4832 

5206 

5580     5953 

6326 

6699 

7071 

7443 

7815 

373 

7 

8186 

8557 

8928 

9298     9668 

0038 

0407 

f)7T< 

1145 

1514 

370 

8 

071882 

2250 

2617 

2985     3352 

3718 

4085 

\Jt  i  O 

4451 

4816 

5182 

366 

9 

5547 

5912 

6276 

6640     7004 

7368 

7731 

8094 

8457 

8819 

363 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

395 
394 

39.5 
39.4 

79.0 

78.8 

118.5 
118.2 

158.0 
157.6 

197.5 
197.0 

237 
236 

.0 
.4 

276.5 
275.8 

316.0 
315.2 

355.5 
354.6 

393 

39.3 

78.6 

11 

7.9 

157.2 

196.5 

235 

.8 

2 

75.1 

314.4 

353.7 

392 

39.2 

78.4 

11 

7.6 

156.8 

196.0 

235 

.2 

"  274.4 

313.6 

352.8 

391 

39.1 

78.2 

11 

7.3 

156.4 

195.5 

234 

.6 

2 

73.7 

312.8 

351.9 

390 

39.0 

78.0 

11 

7.0 

156.0 

195.0 

234 

.0 

2 

73.0 

312.0 

351.0 

389 

38.9 

77.8 

116.7 

155.6 

194.5 

233.4 

272.3 

311.2 

350.1 

388 

38.8 

77.6 

11 

6.4 

155.2 

194.0 

232 

.8 

2 

71.6 

310.4 

349.2 

387 

38.7 

77.4 

116.1 

154.8 

193.5 

232.2 

270.9 

309.6 

348.3 

386 

38.6 

77.2 

11 

5.8 

154.4 

193.0 

231 

.6 

2 

70.2 

308.8 

347.4 

385 

38.5 

77.0 

115.5 

154.0 

192.5 

231 

.0 

269.5 

308.0 

346.5 

384 

38.4 

76.8 

115.2 

153.6 

192.0 

230.4 

268.8 

307.2 

345.6 

383 

38.3 

76.6 

11 

4.9 

153.2 

191.5 

229 

.8 

2 

68.1 

306.4 

344.7 

382 

38.2 

76.4 

114.6 

152.8 

191.0 

229.2 

267.4 

305.6 

343.8 

381 

38.1 

76.2 

11 

4.3 

152.4 

190.5 

228 

.6 

2 

66.7 

304.8 

342.9 

38C 

38.0 

76.  C 

11 

4.0 

152.0 

190.0 

228 

.0 

2 

66.0 

304.0 

342.0 

379 

37.9 

75.8 

113.7 

151.6 

189.5 

227.4 

265.3 

303.2 

341.1 

378 

37.8 

75.6 

11 

3.4 

151.2 

189.0 

226 

.8 

2 

64.6 

302.4 

340.2 

377 

37.7 

75.4 

11 

3.1 

150.8 

188.5 

226 

.2 

2 

63.9 

301.  e 

339.3 

376 

37.6 

75.2 

112.8 

150.4 

188.0 

225.6 

263.2 

300.8 

338.4 

375 

37.5 

75.0 

112.5 

150.0 

187.5 

225.0 

262.5 

300.0 

337.5 

374 

37.4 

74.8 

112.2 

149.6 

187.0 

224.4 

261.8 

299.2 

336.6 

373 

37.3 

74.  e 

111.9 

149.2 

186.5 

223.8 

261.1 

298.4 

335.7 

37$ 

1 

37.2 

74.4 

1 

11 

1.6 

148.8 

186.0 

223 

.2 

2 

60.4 

297.6 

334.8 

371 

37.1 

74.2 

111.3 

148.4 

185.5 

222.6 

259.7 

296.8 

333  9 

37C 

) 

37.0 

74.  ( 

) 

11 

1.0 

148.0 

185.0 

222 

.0 

2 

59.0 

296.0 

333^0 

36< 

) 

36.9 

73.  * 

i 

11 

0.7 

147.6 

184.5 

221 

.4 

2 

58.3 

295.2 

332.1 

36£ 

; 

36.8 

73.6 

110.4 

147.2 

184.0 

220.8 

257.6 

294.4 

331.2 

S6r 

36.7 

73.4 

[ 

11 

0.1 

146.8 

183.5 

22C 

.2 

2 

56.9 

293.6 

330.3 

36( 

5 

36.6 

73.$ 

\ 

1( 

)9.8 

146.4 

183.0 

21S 

.6 

2 

56.2 

292.8 

329.4 

S65 

36.5 

73.0 

109.5 

146.0 

182.5 

219.0 

255.7 

292.0 

328.5 

364 

36.4 

72.? 

\ 

109.2 

145.6 

182.0 

218.4 

254.8 

291.2 

327.6 

1 

36.3 

72.  e 

5 

1( 

)8.9 

145.2 

181.5 

217 

.8 

2 

54.1 

290.4 

326.7 

36$ 

! 

36.2 

72.4 

( 

1( 

)8.6 

•   144.8 

181.0 

217 

.2 

2 

53.4 

289.6 

325.8 

361 

36.1 

72.2 

108.3 

144.4 

180.5 

216.6 

252.7 

288.8 

324.9 

360 

36.0 

72.0 

108.0 

144.0 

180.0 

216.0 

252.0 

288.0 

324.0 

35< 

) 

35.9 

71.1 

] 

1( 

)7.7 

143.6 

179.5 

215 

.4 

2 

51.3 

287.2 

323.1 

3& 

J 

35.8 

71.  ( 

> 

1( 

)7.4 

143.2 

179.0 

214 

.8 

2 

50.6 

286.4 

322.2 

357 

35.7 

71.4 

107.1 

142.8 

178.5 

214 

.2 

249.9 

285.6 

321.3 

85 

> 

35.6 

71.2 

106.8 

142.4 

178.0 

213.6 

249.2 

284.8 

320.4 

150 


TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  120  L.  079.] 

[No.  134  L.  130. 

N. 

0 

1 

2 

3 

4  |   5 

6 

7 

S 

9 

Diff. 

I 

120 

079181  y&43 

9904 

0266 

0626 

0987 

1347 

1707 

2007 

2426  j   300 

1 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

2 
g 

6360 
9905 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

0258 

0611 

0963 

1315 

1667 

2018 

2370 

2721 

3071 

352 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

5 

6910 

7257 

7004 

7951 

8298   8644 

8990 

9335 

9681 



j 

0026 

346 

6 

100371 

0715   1059 

1403 

1747 

2091 

2434 

2777 

3119 

3402 

343 

7 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

341 

8    7210 

7549 

7888 

8227 

8565   8903  9241 

9579 

9916 

0253 

338 

9  110590 

0926 

1363 

1599 

1934   2270 

2005 

2940 

3275 

3609 

335 

130    3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

333 

I     79171 

7603 

7934 

8265 

8595 

8926 

9.250 

9586 

9915 

0245 

330 

2  120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9090 



13 

0012    323 

' 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

355 

35.5 

71.0 

106.5 

142.0 

177.5 

213.0 

248.5 

284.0 

319.5 

354 

35.4 

70.8 

106.2 

141.6 

177.0 

212.4 

247.8 

283.2 

318.6 

353 

35.3 

70.6 

105 

.9 

1 

41.2 

176.5 

211. 

8 

247.1 

282.4 

317.7 

352 

35.2 

70.4 

105.6 

140.8 

176.0 

211.2 

246.4 

281.6 

316.8 

351 

35.1 

70.2 

105 

.3 

1 

40.4 

175.5 

210. 

0 

245.7 

280.8 

315.9 

350 

35.0 

70.0 

105.0 

140.0 

175.0 

210.0 

245.0 

280.0 

315.0 

349 

34.9 

69.8 

104 

.7 

1 

39.6 

174.5 

209. 

4 

244.3 

279.2 

314.1 

348 

34.8 

69.6 

104.4 

139.2 

174.0 

208.8 

243.6 

278.4 

313.2 

347 

34.7 

69.4 

104 

.1 

1 

38.8 

173.5 

208. 

2 

242.9 

277.6 

312.3 

346 

34.6 

69.2 

103.8 

138.4 

173.0 

207.6 

242.2 

276.8 

311.4 

345 

34.5 

69.0 

103.5 

138.0 

172.5 

207.0 

241.5 

276.0 

310.5 

344 

34.4 

68.8 

103 

.2 

1 

37.6 

172.0 

206. 

4 

240.8 

275.2 

309.6 

343 

34.3 

68.6 

102.9 

137.2 

171.5 

205. 

8 

240.1 

274.4 

308.7 

342 

34.2 

68.4 

102 

.6 

1 

36.8 

171.0 

205 

239.4 

273.6 

307.8 

341 

34.1 

68.2 

102 

.3 

1 

36.4 

170.5 

204. 

5 

238.7 

272.8 

306.9 

340 

34.0 

68.0 

102.0 

136.0 

170.0 

204.0 

238.0 

272.0 

306.0 

339 

33.9 

67.8 

101 

.7 

1 

35.6 

169.5 

203. 

4 

237.3 

271.2 

305.1 

338 

33.8 

67.6 

101 

.4 

135.2 

169.0 

202.8 

236.6 

270.4 

304.2 

337 

33.7 

67.4 

101 

.1 

1 

34.8 

168.5 

202. 

235.9 

269.6 

303.3 

336 

33.6 

67.2 

100.8 

134.4 

168.0 

201. 

5 

235.2 

268.8 

302.4 

335 

83.5 

67.0 

100 

.5 

134.0 

167.5 

201.0 

234.5 

268.0 

301.5 

334 

as.  4 

66.8 

100 

.2 

1 

33.6 

167.0 

200. 

4 

233.8 

267.2 

300.6 

333 

as.3 

66.6 

99 

.9 

ias.2 

166.5 

199. 

3 

233.1 

266.4 

299.7 

332 

33.2 

66.4 

99 

.6 

i 

32.8 

166.0 

199. 

2 

232.4 

265.6 

298.8 

331 

as.i 

66.2 

99.3 

132.4 

165.5 

198. 

3 

231.7 

264.8 

297.9 

aso 

33.0 

66.0 

99 

.0 

] 

32.0 

165.0 

198. 

} 

231.0 

264.0 

297.0 

329 

32.9 

65.8 

98 

.7 

31.6 

164.5 

197.  < 

t 

230.3 

263.2 

296.1 

328 

32.8 

65.6 

98 

.4 

131.2 

164.0 

196. 

3 

229.6 

262.4 

295.2 

327 

32.7 

65.4 

98 

.1 

1 

30.8 

103.5   196. 

2 

228.9 

201.6 

294.3 

326 

32.6 

05.2 

97 

.8 

130.4 

103.0   195.6 

228.2 

260.8 

293.4 

325 

32.5 

65.0 

97.5 

130.0 

162.5   195.0 

227.5 

260.0 

292.5 

324 

32.4 

64.8 

97 

.2 

129.6 

162.0   194.4 

226.8 

259.2 

291.6 

323 

32.3 

64.6 

96 

.9 

1 

29.2 

161.5   193.  J 

J 

226.1 

258.4 

290.7 

322 

32.2 

64.4 

96 

.6 

128.8 

161.0   193.2 

225.4 

257.6 

289.8 

151 


TABLE    XI. — LOGARITHMS   OF   KtlMBERS. 


No.  135  L.  130.] 

[No.  149  L.  175. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8         9 

Diff. 

135 

130334 

0655 

0977 

1298 

1619 

1939     2260 

2580 

2900     3219 

321 

6 

3539 

3858 

4177 

4496 

4814 

5133  '  5451 

5769 

6086     6403 

318 

7 
g 

6721 

9879 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249     9564 

316 

0194 

0508 

0822 

1136 

1450 

1763 

2076 

2389     2702 

314 

9 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507     5818 

311 

140 

6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603     8911 

309 

9219 

9527 

9835 

0142 

0449 

0756 

1063 

1370 

1676     1982 

307 

2 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728     5032 

305 

3 

4 

5336 

8362 

5640 
8664 

5943 
8965 

6246 
9266 

6549 
9567 

6852 
9868 

7154 

7457 

7759     8001 

303 

0168 

04G9 

0769     1068 

301 

5 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758     4055 

299 

6 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726     7022 

297 

7 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674     9968 

295 

8 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603     2895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512     5802 

291 

PROPORTIONAL  PARTS. 

Diff. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

321 

32.1 

64.2 

96.3 

128.4 

160.5 

192 

6 

224.7 

256.8 

288.9 

320 

32.0 

64.0 

96.0 

128.0 

160.0 

192. 

0 

224.0 

256.0 

288.0 

319 

31.9 

63.8 

95 

.7 

127.6 

159.5 

191 

4 

2£ 

3.3 

255.2 

287.1 

318 

31.8 

63.6 

95.4 

127.2 

159.0 

190 

8 

222.6 

254.4 

286.2 

317 

31.7 

63.4 

95 

.1 

126.8 

158.5 

190 

2 

2S 

1.9 

253.6 

285.3 

316 

31.6 

63.2 

94 

.8 

126.4 

158.0 

189 

6 

2$ 

1.2 

252.8 

284.4 

315 

31.5 

63.0 

94 

.5 

126.0 

157.5 

189 

0 

220.5 

252.0 

283.5 

314 

31.4 

62.8 

94 

.2 

125.6 

157.0 

188 

4 

21 

9.8 

251.2 

282.6 

313 

31.3 

62.6 

93 

9 

125.2 

156.5 

187 

8 

219.1 

250.4 

281.7 

312 

31.2 

62.4 

93 

.6 

124.8 

156.0 

187 

2 

218.4 

249.6 

280.8 

311 

31.1 

62.2 

93 

.3 

124.4 

155.5 

186 

6 

217.7 

248.8 

279.9 

310 

31.0 

62.0 

93 

.0 

124.0 

155.0 

186 

0 

21 

7.0 

248.0 

279.0 

309 

30.9 

61.8 

92.7 

123.6 

154.5 

185 

4 

216.3 

247.2 

278.1 

308 

30.8 

61.6 

92 

.4 

123.2 

154.0 

184 

8 

21 

5.6 

246.4 

277.2 

307 

30.7 

61.4 

92 

.1 

L£2  8 

153.5 

184 

0 

214.9 

245.6 

276.3 

306 

30.6 

61.2 

91 

.8 

122  .'4 

153.0 

183 

6 

21 

4.2 

244.8 

275.4 

305 

30.5 

61.0 

91 

.5 

122.0 

152.5 

183 

0 

21 

3.5 

244.0 

274  ,5 

304 

30.4 

60.8 

91 

.2 

121.6 

152.0 

182 

4 

212.8 

243.2 

273.6 

303 

30.3 

60.6 

90 

.9 

121.2 

151.5 

181 

8 

21 

2.1 

242.4 

272.7 

302 

30.2 

60.4 

90 

.6 

120.8 

151.0 

181 

2 

211.4 

241.6 

271.8 

301 

30.1 

60.2 

90 

.3 

120.4 

150.5 

180.6 

210.7 

240.8 

270.9 

300 

30.0 

60.0 

90 

.0 

120.0 

150.0 

180.0 

210.0 

240.0 

270.0 

299 

29.9 

59.8 

89 

.7 

119.6 

149.5 

179 

4 

2C 

)9.3 

239.2 

269.1 

298 

29.8 

59.6 

89 

.4 

119.2 

149.0 

178.8 

208.6 

238.4 

268.2 

297 

29.7 

59.4 

89 

.1 

118.8 

148.5 

178 

2 

2( 

)7.9 

237.6 

267.3 

296 

29.6 

59.2 

88 

.8 

118.4 

148.0 

177 

6 

207.2 

236.8 

266.4 

295 

29.5 

59.0 

88 

.5 

118.0 

147.5 

177 

0 

2( 

)6.5 

236.0 

265.5 

294 

29.4 

58.8 

88.2 

117.6 

147.0 

176 

4 

205.8       235.2 

264.6 

293 

29.3 

58.6 

87 

.9 

117.2 

146.5 

175 

8 

2C 

)5.1 

234.4 

263.7 

292 

29.2 

58.4 

87 

.6 

116.8 

146.0 

175 

2 

204.4 

233.6 

262.8 

291 

29.1 

58.2' 

87 

.3 

116.4 

145.5 

174 

6 

2( 

)3.7 

232.8 

261.9 

290 

29.0 

58.0 

87.0 

116.0       145.0 

174 

0 

203.0 

232.0 

261.0 

289 

28.9 

57.8 

86 

.7 

115.6 

144.5 

173 

4 

2C 

2.3 

231.2 

260.1 

288 

28.8 

57.6 

86 

.4 

115.2 

144.0 

172.8 

201.6 

230.4 

259.2 

287 

28.7 

57.4 

86 

.1 

114.8 

143.5 

172 

2 

2C 

>0.9 

229.6 

258.3 

286 

28.6 

57.2 

85 

.8 

114.4 

143.0 

171 

6 

200.2 

228.8 

257.4 

152 


TABLE   XI. — LOGARITHMS    OF    NUMBERS. 


No.  150  L.  176.] 

[No.  169  L.  230. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

150~ 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

8977 

9264 

9552 

9839 

0126 

0413 

0699 

0986 

19^9 

tf&R 

007 

2 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

mot 

285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956  7239 

283 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

AAK-J 

931 

5 

190332 

0812 

0892 

1171 

1451 

1730 

2010 

2289 

—  —  —  UUO1 

2567  2846 

SBJL 

279 

6 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346  5623 

378 

7 

5900 

6176 

6453  6729 

7005 

7281 

7'556 

7832 

8107  8382 

276 

g 

8657 

8932 

9206 

9481 

9755 

0029 

0303 

0577 

0850   1124 

97  A. 

9 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577  3848 

iCn 

272 

160 

4120* 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

1 

6826 

7096 

7365 

7634 

7904 

8173 

8441  8710 

8979 

9247 

269 

2 

9515 

9783 

0051 

0319 

0586 

0853 

1121  |  1388 

1654  '  1QO1 

267 

3 

212188 

2454 

2720 

2986 

3252 

3518 

3783  4049 

4314 

4579 

266 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

7 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

9 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

23 

0193 

256 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

285 

28.5 

57.0 

85.5 

114.0 

142.5 

171.0 

199.5 

228.0 

256.5 

284 

28.4 

56.8 

85.2 

113.6 

142.0 

170.4 

198.8 

227.2 

255.6 

283 

28.3 

56.6 

84.9 

113.2 

141.5 

169.8 

198.1 

226.4 

254.7 

282 

28.2 

56.4 

84.6 

112.8 

141.0 

169.2 

197.4 

225.6 

253.8 

281 

28.1 

56.2 

84.3 

112  4 

140.5 

168.6 

196.7 

224.8 

252.9 

280 

28.0 

56.0 

84.0 

112.0 

140.0 

168.0 

196.0 

224.0 

252.0 

279 

27.9 

55.8 

83.7 

111.6 

139.5 

167.4 

195.3 

223.2 

251.1 

278 

27.8 

55.6 

83.4 

111.2 

139.0 

166.8 

194.6 

222.4 

250.2 

277 

27.7 

55.4 

83.1 

110.8 

138.5 

166.2 

193.9 

221.6 

249.3 

276 

27.6 

55.2 

82.8 

110.4 

138.0 

165.6 

193.2 

220.8 

248.4 

275 

27.5 

55.0 

82.5 

110.0 

137.5 

165.0 

192.5 

220.0 

247.5 

274 

27.4 

54.8 

82.2 

109.6 

137.0 

164.4 

191.8 

219.2 

246.6 

273 

27.3 

54.6 

81.9 

109.2 

136.5 

163.8 

191.1 

218.4 

245.7 

272 

27.2 

54.4 

81.6 

108.8 

136.0 

163.2 

190.4 

217.6 

244.8 

271 

27.1 

54.2 

81.3 

108.4 

135.5 

162.6 

189.7 

216.8 

243.9 

270 

27.0 

54.0 

81.0 

108.0 

135.0 

162.0 

189.0 

216.0 

243.0 

269 

26.9 

53.  g 

80.7 

107.6 

134.5 

161.4 

188.3 

215.2 

242.1 

268 

26.8 

53.6 

80.4 

107.2 

134.0 

160.8 

187.6 

214.4 

241.2 

267 

26.7 

53.4 

80.1 

106.8 

133.5 

160.2 

186.9 

213.6 

240.3 

266 

26.6 

53.2 

79.8 

106.4 

133.0 

159.6 

186.2 

212.8 

239.4 

265 

26.5 

53.0 

79.5 

106.0 

132.5 

159.0 

185.5 

212.0 

238.5 

264 

26.4 

52.  g 

79.2 

105.6 

132.0 

158.4 

184.8 

211.2 

237.6 

263 

26.3 

52.  € 

78.9 

105.2 

131.5 

157.8 

184.1 

210.4 

236.7 

262 

26.2 

52.4 

78.6 

104.8 

131.0 

157.2 

183.4 

209.6 

235.8 

261 

26.1 

52.2 

78.3 

104.4 

130.5 

156.6 

182.7 

208.8 

234.9 

260 

26.0 

52.0 

78.0 

104.0 

130.0 

156.0 

182.0 

208.0 

234.0 

259 

25.9 

61.1 

77.7 

103.6 

129.5 

155.4 

181.3 

207.2 

233.1 

258 

25.8 

51.  e 

> 

77.4 

103.2 

129.0 

154.8 

180.6 

206.4 

232.2 

257 

25.7 

51.4 

102.8 

128.5 

154.2 

179.9 

205.6 

231.3 

256 

25.6 

51.  S 

5 

76  '.8 

102.4 

128.0 

153.6 

179.2 

204.8 

230.4 

255 

25.5 

51.0 

76.5 

102.0 

1£7.5 

153.0 

178.5 

204.0 

229.5 

153 


TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  170  L.  230.]                                  [No.  189  L.  278. 

N. 

0 

1 

2 

8 

4 

6 

6 

7 

8 

9 

Diff. 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742   255 

1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

0050 

0300 

OKA 

4 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

/<iOU 

249 

5 

3038 

3286 

35:34 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

6 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

0176 

245 

8 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

243 

9 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

1 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

g 

9513 

9746 

9980 

0213 

0446 

0679 

0912 

1144 

1377 

1609 

oqq 

7 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

600 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

9 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

255 
254 

25.5 

25.4 

51.0 

50.8 

76.5 
76.2 

102.0 
101.6 

127.5 
127.0 

153.0 
152.4 

178.5 

177.8 

204.0 
203.2 

229.5 

228.6 

253 

25.3 

50.6 

75.9 

101.2 

126.5 

151.8 

177.1 

202.4 

227.7 

252 

25.2 

50.4 

75.6 

100.8 

126.0 

151.2 

176.4 

201.6 

226.8 

251 

25.1 

50.2 

75.3 

100.4 

125.5 

150.6 

175.7 

200.8 

225.9 

250 

25  0 

50.0 

75.0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

249 

24.9 

49.8 

74.7 

99.6 

124.5 

149.4 

174.3 

199.2 

224.1 

248 

24.8 

49.6 

74.4 

99.2 

124.0 

148.8 

173.6 

198.4 

223.2 

247 

24.7 

49.4 

74.1 

98.8 

123.5 

148.2 

172.9 

197.6 

222.3 

246 

24.6 

49.2 

73.8 

98.4 

123.0 

147.6 

172.2 

196.8 

221.4 

245 

24.5 

49.0 

73.5 

98.0 

122.5 

147.0 

171.5 

196.0 

220.5 

244 

24.4 

48.8 

73.2 

97.6 

122.0 

146.4 

170.8 

195.2 

219.6 

243 

24.3 

48.6 

72.9 

97.2 

121.5 

145.8 

170.1 

194.4 

218.7 

242 

24.2 

48.4 

72.6 

96.8 

121.0 

145.2 

169.4 

193.6 

217.8 

241 

24.1 

48.2 

72.3 

96.4 

120.5 

144.6 

168.7 

192.8 

216.9 

240 

24.0 

48.0 

72.0 

96.0 

120.0 

144.0 

168.0 

192.0 

216.0 

239 

23.9 

47.8 

71.7 

95.6 

119.5 

143.4 

167.3 

191.2 

215.1 

238 

23.8 

47.6 

71.4 

95.2 

119.0 

142.8 

166.6 

190.4 

214.2 

237 

23.7 

47.4 

71.1 

94.8 

118.5 

142.2 

165.9 

189.6 

213.3 

236 

23.6 

47.2 

70.8 

94.4 

118.0 

141.6 

165.2 

188.8 

212.4 

235 

23.5 

47.0 

70.5 

94.0 

117.5 

141.0 

164.5 

188.0 

211.5 

234 

23.4 

46.8 

70.2 

93.6 

117.0 

140.4 

163.8 

187.2 

210.6 

233 

23.3 

46.6 

69.9 

93.2 

116.5 

139.8 

163.1 

186.4 

209.7 

232 

23.2 

46.4 

69.6 

92.8 

116.0 

139.2 

162.4 

185.6 

208.8 

231 

23.1 

46.2 

69.3 

92.4 

115.5 

138.6 

161.7 

184.8 

207.9 

230 

23.0 

46.0 

69.0 

92.0 

115.0 

138.0 

161.0 

184.0 

207.0 

229 

22.9 

45.8 

68.7 

91.6 

114.5 

137.4 

160.3 

183.2 

206.1 

228 

22.8 

45.6 

68.4 

91.2 

114.0 

136.8 

159.6 

182.4 

205.2 

227 

22.7 

45.4 

68.1 

90.8 

113.5 

136.2 

158.9 

181.6 

204.3 

226 

22.6 

45.2 

67.8 

90.4 

113.0 

135.6 

158.2 

180.8 

203.4 

154 


TABLE    XI. — LOGAEITHMS    OF   NUMBERS. 


No.  190  L.  278.]                                                                                   [No.  214  L.  332. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

190 

278754 

8982 

9211 

9439 

9667 

9895 

I  Ki  li' 

ftlOQ 

rvoe-j 

nK»"Q 

OOQ 

1 

281033 

1261 

1488 

1715 

1942 

2169 

&m 

a  ,-j-j 

2849 

3075 

227 

2 

3301 

3527 

3753 

3979 

4205 

4431 

4056 

4882 

5107 

5332 

226 

3 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

5 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

20:34 

222 

6 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

821 

7 

44G6 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

68&4 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

g 

8853 

9071 

9289 

9507 

9725 

9943 

0161 

0378 

0595 

0813 

218 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

1 

3196 

*3412 

36.<28 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

3 

7496 

7710 

7924 

8137 

8351 

8564 

877'8 

8991 

9204 

9417 

213 

4 

9630 

9843 

0056 

0268 

0481 

0693 

0906 

1118 

1330 

1542 

212 

5 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3415 

3656 

211 

6 

3867 

4078 

4289 

4499 

4710 

!  4920 

5130 

5340 

5551 

5760 

210 

7 

5970 

6180 

6390 

6599 

6809 

7018 

72:37 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

0354 

0562 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

1 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

0008 

0211 

203 

4 

330414 

0617 

0819 

1022  I  1225 

1427 

1630 

1&32 

2034 

2236 

202 

PROPORTIONAL  PARTS. 

Biff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

225 

22.5 

45.0 

67.5 

90.0 

112.5 

135.0 

157.5 

180.0 

202.5 

224 

22.4 

44,.  8 

67.2 

89.6 

112.0 

134.4 

156.8 

179.2 

201.6 

223 

22.3 

44.6 

66.9 

89.2 

111.5 

133.8 

156.1 

178.4     200.7 

222 

22.2 

44.4 

66.6 

88.8 

111.0 

1,33.2 

155.4 

177.6     199.8 

221 

22.1 

44.2 

66.3 

88.4 

110.5 

132.6 

154.7 

176.8     198.9 

220 

22.0 

44.0 

66.0 

88.0 

110.0 

132.0 

154.0 

176.0 

198.0 

219 

21  9 

43  8 

65.7 

87.6 

109.5 

131.4 

153.3 

175.2 

197.1 

218 

21.8 

43.6 

65.4 

87.2 

109.0 

130.8 

152.6 

174.4 

196.2 

217 

21.7 

43.4 

65.1 

86.8 

108.5 

130.2 

151.9 

173.6 

195.3 

216 

21.6 

43.2 

64.8 

86.4 

108.0 

129.6 

151.2 

172.8 

194.4 

215 

21.5 

43.0 

64.5 

86.0 

107.5 

129.0 

150.5 

172.0 

193.5 

214 

21.4 

42.8 

64.2 

85.6 

107.0 

128.4 

149.8 

171.2 

198.fi 

213 

21.3 

42.6 

63.9 

85.2 

106.5 

127.8 

149.1 

170.4 

191.7 

212 

21.2 

42.4 

63.6 

84.8 

106.0 

127.2 

148.4 

169.6 

190.8 

211 

21.1 

42.2 

63.3 

84.4 

105.5 

126.6 

147.7 

168.8 

180.9 

210 

21.0 

42.0 

63.0 

84.0 

105.0 

126.0 

147.0 

168.0 

189.0 

209 

20.9 

41.8 

62.7 

83.6 

104.5 

125.4 

146.3 

167.2 

188.1 

208 

20.8 

41.6 

62.4 

&3.2 

104.0 

124.8 

145.6 

166  4 

187  J 

207 

20.7 

41.4 

62.1 

82.8 

103.5 

124.2 

144.9 

165.6 

186.3 

206 
205 
204 
203 
202 

20.6 
20.5 
20.4 
20.3 
20.2 

41.2 
44.0 
40.8 
40.6 
40.4 

61.8 
C1.5 
61.2 
60.9 
60.6 

82.4 
82.0 
81.6 
81.2 
'X).8 

103.0 
102.5 
102.0 
101.5 
101.0 

123.6 
123.0 
122.4 
121.8 
121.2 

144.2 
143.5 
142.8 
142.1 
141.4 

164.8 
164.0 
163.2 
162.4 
161.6 

185.4 
184.5 

183.6 
182.7 
181.8 

155 


TABLE    XI. — LOGARITHMS   OF   NUMBERS. 


No.  215  L.  332.]                                  [No.  239  L.  380. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

215 

332438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202 

6 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

g 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

0047 

0246 

199 

9 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

198 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

2 

6353 

6549 

6?'44 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

g 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

-  0054 

194 

4 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

193 

5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

&532 

3724 

3916 

193 

6 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

-  191 

8 

7935 
9835 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

0025 

0215 

0404 

0593 

0783 

097° 

1161 

1350 

1539 

189 

230 

361728 

1917 

2105 

2294- 

2482 

2671 

2859 

3048 

3236 

3424 

188 

1 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

977° 

9958 

0143 

0328 

0513 

0698 

0883 

18*4 

5 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

43H2 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

g 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

38 

0030 

181 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

202 
201 

20.2 
20.1 

40.4 
40.2 

60.6 
60.3 

80.8 
80.4 

101.0 
100.5 

121.2 
120.6 

141.4 
140.7 

161.6 
160.8 

181.8 
180.9 

200 

20.0 

40.0 

60.0 

80.0 

100.0 

120.0 

140.0 

160.0 

180.0 

199 

19.9 

39.8 

59.7 

79.6 

99.5 

119.4 

139.3 

159.2 

179.1 

198 

19.8 

39.6 

59.4 

79.2 

99.0 

118.8 

138.6 

158.4 

178.2 

197 

19.7 

39.4 

59.1 

78.8 

98.5 

118.2 

137.9 

157.6 

177.3 

196 

19  6 

39.2 

58.8 

78.4 

98  0 

117.6 

137.2 

156.8 

176.4 

195 

19.5 

39.0 

58.5 

78.0 

97.5 

117.0 

136.5 

156.0 

175.5 

194 

19.4 

38.8 

58.2 

77.6 

97.0 

116.4 

135.8 

155.2 

174.6 

193 

19  3 

38.6 

57.9 

77.2 

96.5 

115.8 

135.1 

154.4 

173.7 

192 

19  2 

38.4 

57.6 

76.8 

96.0 

115.2 

134.4 

153.6 

172.8 

191 

19.1 

38.2 

57.3 

76.4 

95.5 

114.6 

133.7 

152.8 

171.9 

190 

19.0 

38.0 

57.0 

76.0 

95.0 

114.0 

133.0 

152.0 

171.0 

189 

18.9 

37.8 

56.7 

75.6 

94  5 

113.4 

132.3 

151.2 

170.1 

188 

18.8 

37.6 

56.4 

75.2 

94.0 

112.8 

131.6 

150.4 

169.2 

187 

18.7 

37  4 

56.1 

74.8 

93.5 

112.2 

130.9 

149.6 

168.3 

186 

18.6 

37.2 

55.8 

74.4 

93.0 

111.6 

130.2 

148.8 

167.4 

185 

18  5 

37.0 

55  5 

74.0 

92.5 

111.0 

129.5 

148.0 

166.5 

184 

18  4 

36  8 

55.2 

73.6 

92.0 

110.4 

128.8 

147.2 

165.6 

183 

18  3 

36.6 

54.9 

73  2 

91.5 

109.8 

128.1 

146.4 

164.7 

182 

18  2 

36.4 

54  6 

72.8 

91.0 

109  2 

127.4 

145.6 

163.8 

181 

18  1 

36  2 

54.3 

724     90.5 

108  6 

126.7 

144.8 

162.9 

180 

18  0 

36.0 

54  0 

72.0    90  0 

108.0 

126.0 

144.0 

162.0 

179 

17.9 

35.8 

53.7 

71.6    89.5 

107.4 

125.3 

143.2 

161.1 

156 


TABLE    XI. — LOOAKITHMS    OF    Xt'MBERS. 


No.  240  L.  380.] 

INo.  269  L.  431. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

181 

1 

2017 

2197 

2377 

25. 

)7 

2737 

2917 

3097 

3 

277 

345 

6 

3636 

180 

2 

3815 

3995 

4174 

43, 

33 

4533 

4712 

4891 

5 

070 

524 

> 

5428 

179 

3 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

4 

7390 

7568 

7746 

7$ 

U 

8101 

8279 

8456 

8 

634 

881 

1 

8989 

178 

5 

9166 

9343 

9520 

90( 

)8 

9875 

0051 

ft9OQ 

f\ 

4(\K 

-... 

6 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

(J-HJO 

2169 

058« 
2345 

()  1  90 

XJ.VJ1 

17  i 

176 

7 

2697 

2873 

3048 

32S 

U 

3400 

3575 

3751 

3 

926 

410 

1 

4277 

176 

8 

4452 

4027 

4802 

49 

"7 

5152 

5326 

5501 

5 

676 

585 

0 

0025 

175 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7 

419 

7592 

7766 

174 

250 

7940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

1 

9674 

9847 

0020 

01 

V} 

0365 

_„_„ 

ft 

ooo 

oi>»i 

•*T> 

2 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

Uooo 
2605 

105o 
2777 

2949 

178 

172 

3 

3121 

3292 

3464 

3635 

.3807 

3978 

4149 

4320 

4492 

4663 

171 

4 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

5 

6540 

6710 

6881 

70, 

U 

7221 

7391 

7561 

731 

790 

1 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

109 

7 

9933 

0102 

0271 

0440 

0609 

0777 

0946 

1114 

1283 

1451 

169 

8 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

4973 

5140 

5307 

5474 

5641 

5808 

5974 

0141 

6308 

6474 

167 

1 

6641 

6807 

6973 

71' 

*9 

7306 

7472 

7638 

804 

797 

0 

8135 

166 

2 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

3 

9956 

0121 

0286 

04. 

^1 

0616 

0781 

0945 

•i 

i1A 

197 

K 

14VJ 

165 

4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

iLL\J 

27.54 

l*lu 

2918 

3082 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

5045 

5208 

53 

"1 

5534 

5697 

5860 

6 

023 

618 

0 

6349 

163 

7 

6511 

6674 

6836 

69< 

)9 

7161 

!  7324 

7486 

7 

648 

781 

1 

7973 

162 

8 
g 

8135 
9752 

8297 
9914 

8459 

8621 

8783 

|  8944 

9106 

9268 

9429 

9591 

162 

43 

0075  0236  0398 

!  0559 

0720 

0881 

1042 

1203 

161 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3 

4 

5 

6 

7 

8 

9 

178 

17.8 

35.6 

53.4 

71.2 

89.0 

106.8 

124.6 

142.4 

160.2 

177 

17.7 

35.4 

53.1 

70.8 

88.5 

106.2 

123.9 

141.6 

159.3 

176 

17.6 

35.2 

52.8 

70.4 

88.0 

105.6 

123.2 

140.8 

158  I 

175 

17.5 

35.0 

52.5 

70.0 

87.5 

105.0 

122.5 

140.0 

174 

17.4 

34.8 

52.2 

69.6 

87.0 

104.4 

121.8 

139.2 

156.6 

173 

17.3 

34.6 

51.9 

69.2 

86.5 

103.8 

121.1 

138.4 

155.7 

172 

17.2 

34.4 

51.6 

68.8 

86.0 

103.2 

1.20.4 

137.6 

154.8 

171 

17.1 

34.2 

51.3 

68.4 

85.5 

102.6 

119.7 

136.8 

153.9 

170 

17.0 

34.0 

51.0 

68.0 

85.0 

102.0 

119.0 

136.0 

153.0 

169 

16.9 

33.8 

50.7 

67.6 

84.5 

101.4 

118.3 

13.-).  2 

152.1 

168 

16.8 

33.6 

50.4 

67.2 

84.0 

100.8 

117.6 

134.4 

151.2 

167 

16.7 

as.  4 

50.1 

66.8 

83.5 

100.2 

116.9 

133.6 

150.3 

166 

16.6 

33.2 

49.8 

66.4 

83.0 

99.6 

116.2 

132.8 

149.4 

165 

16.5 

33.0 

49.5 

66.0 

82.5 

99.0 

115.5 

132.0 

148.5 

164 

16.4 

32.8 

49.2 

65.6 

82.0 

98.4 

114.8 

131.2 

147.6 

163 

16.3 

32.6 

48.9 

65.2 

81.5 

97.8 

114.1 

130.4 

no.  7 

162 

16.2 

32.4 

48.5 

64.8 

81.0 

97.2 

113.4 

129.6 

161 

16.1 

32.2 

48.3 

64.4 

80.5 

96.6 

112.7 

128.8 

n  »  .  (j 

157 


TABLE    XI. — LOGARITHMS   OF    NUMBERS. 


No.  270  L.  431.] 

[No.  299  L.  476. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

& 

m 

2649 

2809 

161 

1 

2969 

3130 

8290 

3450 

3610 

3770 

3930 

4( 

)90 

4249 

4409 

160 

2 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7' 

275 

7433 

7592 

159 

4 

7751 

7909 

8067 

8226 

8384 

8542  8701 

8 

359 

9017 

9175 

158 

5 

9333 

9491 

9648 

9806 

9964 

0122 

0279 

o^ 

137 

0594 

0752 

158 

6 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

2323 

157 

7 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3, 

376 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

9 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

155 

1 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

0095 

154 

2 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2i 

359 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5 

310 

6062  ,  6214 

152 

6 

6366 

6518 

6670 

6821 

6973 

7125 

727'6 

7428 

7579 

7731 

152 

7 
g 

7882 
9392 

8033 
9543 

8184 
9694 

8336 
9845 

8487 
9995 

8638 

8789 

8940 

9091 

9242 

151 

0146 

0296 

0% 

i/17 

0597 

0748 

151 

9 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

150 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

150 

1 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4 

J36 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7 

J04 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085  9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

0116 

0263 

0410 

0557 

0704 

0 

asi 

0998 

1145 

147 

6 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5 

235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

161 

16.1 

32.2 

48.3 

64.4 

80.5 

96.6 

112.7 

128.8 

144.9 

160 

16.0 

32.0 

48.0 

64.0 

80.0 

96.0 

112.0 

128.0 

144.0 

159 

15.9 

31.8 

47.7 

63.6 

79.5 

95.4 

111.3 

127.2 

143.1 

158 

15.8 

31.6 

47.4 

63.2 

79.0 

94.8 

110.6 

126.4 

142.2 

157 

15.7 

31.4 

47.1 

62.8 

78.5 

94.2 

109.9 

125.6 

141.3 

156 

15.6 

31.2 

46.8 

62.4 

78.0 

93.6 

109.2 

124.8 

140.4 

155 

15.5 

31.0 

46.5 

62.0 

77.5 

93.0 

108.5 

124.0 

139.5 

154 

15.4 

30.8 

46.2 

61.6 

77.0 

92.4 

107.8 

123.2 

138.6 

153 

15.3 

30.6 

45.9 

61.2 

76.5 

91.8 

107.1 

122.4 

137.7 

152 

15.2 

30.4 

45.6 

60.8 

76.0 

91.2 

106.4 

121.6 

136.8 

151 

15.1 

30.2 

45.3 

60.4 

75.5 

90.6 

105.7 

120.8 

135.9 

150 

15.0 

30.0 

45.0 

60.0 

75.0 

90.0 

105.0 

120.0 

135.0 

149 

14.9 

29.8 

44.7 

59.6 

74.5 

89.4 

104.3 

119.2 

134.1 

148 

14.8 

29.6 

44.4 

59.2 

74.0 

88.8 

103.6 

118.4 

133.2 

147 

14.7 

29.4 

44.1 

58.8 

73.5 

88.2 

102.9 

117.6 

132.3 

146 

14.6 

29.2 

43.8 

58.4 

73.0 

87.6 

102.2 

116.8 

131.4 

145 

14.5 

29.0 

43.5 

58.0 

72.5 

87.0 

101.5 

116.0 

130.5 

144 

14.4 

28.8 

43.2 

57.6 

72.0 

86.4 

100.8 

115.2 

129.6 

143 

14.3 

28.6 

42.9 

57.2 

71.5 

85.8 

100.1 

114.4 

128.7 

142 

14.2 

28.4 

42.6 

56.8 

71.0 

85.2 

99.4 

113.6 

127.8 

141 

14.1 

28.2 

42.3 

56.4 

70.5 

84.6 

98.7 

112.8 

126.9 

140 

14.0 

28.0 

42.0 

56.0 

70.0 

84.0 

98.0 

112.0 

126.0 

158 


TABLE    XI.  — LOGARITHMS    OF    NUMBERS. 


No.  300  L.  477.] 

[No.  339  L.  531. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diflf. 

145 
144 

144 
143 
143 
142 
142 
141 
141 

140 

140 
139 
139 
139 
138 
138 

137 
137 
136 
136 

136 
135 
135 

134 
134 
133 
133 
133 
132 
132 

131 

131 
131 
130 
130 
129 
129 
129 

128 
128 

300 

1 

2 
3 
4 
5 
6 

8 
9 

310 
1 
2 
3 
4 
5 
6 

7 
8 
9 

320 
1 
2 
3 

4 
5 
6 

7 
8 
9 

330 
1 

2 
3 
4 
5 
6 
7 
8 

9 

477121 
8566 

7266 
8711 

7411 

8855 

7555 
8999 

7700 
9143 

7844 
9287 

7989 
9431 

8133 
9575 

8278 
9719 

8422 
9863 

480007 
1443 
2874 
4300 
5721 
7138 
8551 
9958 

0151 
1586 
3016 
4442 

5863 

7280 
8692 

0294 
1729 
3159 
4585 
6005 
7421 
8833 

0438 
1872 
3302 
4727 
6147 
7563 
8974 

0582 
2016 
3445 
4869 
6289 
7704 
9114 

0725 
2159 
3587 
5011 
6430 
7845 
9255 

0869 
2:302 
3730 
5153 
6572 
7986 
9396 

1012 
2445 
3872 
5295 
6714 
8127 
9537 

1156 
2588 
4015 
5437 
C855 
8269 
9077 

1299 
2731 
4157 
5579 
6997 
8410 
9818 

0099 

1502 
2900 
4294 
5683 
7068 
8448 
9824 

0239 

1642 
3040 
4433 

5822 
7206 
8586 
9962 

0380 

1782 
3179 
4572 
5960 
7344 
8724 

0520 

1922 
3319 
4711 
6099 

7483 
8862 

0661 

2062 
3458 
4850 
6238 
7621 
8999 

0801 

2201 
3597 
4989 
6376 
7759 
9137 

0941 

2341 
3737 
5128 
6515 
7897 
9275 

1081 

2481 
3876 
5267 
6653 
8035 
9412 

1222 

2621 
4015 
5406 
0791 
8173 
9550 

491362 
2760 
4155 
5544 
6930 
8311 
9687 

0099 
1470 
2837 
4199 

5557 
6911 
8260 
9606 

0236 
1607 
2973 
4:335 

5693 
7046 
8395 
9740 

0374 
1744 
3109 
4471 

5828 
7181 
8530 
9874 

0511 
1880 
3246 
4607 

5964 
7316 
8664 

0648 
2017- 
3382 
4743 

6099 
7451 
8799 

0785 
2154 
3518 
4878 

6234 
7586 
8934 

0922 
2291 
3655 
5014 

6370 
7721 
9068 

501059 
2427 
3791 

5150 
6505 
7&56 
9203 

1196 
2564 
3927 

5286 
6640 
7991 
9337 

ias3 

2700 
4063 

5421 
6776 
8126 
9471 

0009 
1349 
2684 
4016 
5344 
6668 
7987 

9303 

0615 
1922 
3226 
4526 
5822 
7114 
8402 
9687 

0143 
1482 
2818 
4149 
5476 
6800 
8119 

9434 

0745 
2053 
3356 
4656 
5951 
7243 
8531 
9815 

0277 
1616 
2951 
4282 
5609 
6932 
8251 

9566 

0876 
2183 
3486 
4785 
C081 
7372 
80CO 
9943 

0411 
1750 
£084 
4415 
5741 
7064 
8382 

9697 

1C07 
2314 
3616 
4915 
6210 
7501 
8788 

510545 
1883 
3218 
4548 
5874 
7196 

8514 
9828 

0679 
2017 
3351 
4681 
6006 
7328 

8646 
9959 

0813 
2151 
3484 
4813 
6139 
7460 

8777 

0947 
2284 
3617 
4946 
6271 
7592 

8909 

1081 
2418 
3750 
5079 
6403 
7724 

9040 

1215 
2551 

3883 
5211 
6535 

7855 

9171 

0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 

0221 
1530 
2835 
4136 
5434 
6727 
8016 
9-302 

0353 
1661 
2966 
4266 
5563 
6856 
8145 
9430 

0484 
1792 
3096 
4396 
5693 
6985 
8274 
9559 

521138 
2444 
3746 
5045 
6339 
7630 
8917 

1269 
2575 
3876 
5174 
6469 
7759 
9045 

0072 
1351 

530200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

3 

4 

5 

6 

7 

8 

9 
125.1 

123.3 
122.4 
121.5 
120.6 

11'.'  7 
118.8 
117.9 

117.0 
116.1 

11.  VJ 
111  .:; 

139   13.9 
138   13.8 
137   13.7 
136   13.6 
135   13.5 
134   13.4 
133   13.3 
132   13.2 
131   13.1 
130   13.0 
129   12.9 
128   12.8 
127   12  7 

27.8 
27.6 
27.4 
27.2 
27.0 
26.8 
26.6 
26.4 
26.2 
26.0 
25.8 
25.6 
25.4 

41.7 
41.4 
41.1 
40.8 
40.5 
40.2 
39.9 
39.6 
89.3 
89.0 
38.7 
38.4 
38.1 

55.6 
55.2 
54.8 
54.4 
54.0 
53.6 
53.2 
52.8 
52.4 
52.0 
51.6 
51.2 
50.8 

69.5 
69.0 
68.5 
68.0 
67.5 
67.0 
66.5 
66.0 
65.5 
65.0 
64.5 
64.0 
63.5 

83.4 

82.8 
82.2 
81.6 
81.0 
80.4 
79.8 
79.2 
78.6 
78.0 
77.4 
76.8 
76.2 

97.3 
%..6 
95.9 
95.2 
94.5 
93.8 
93.1 
92.4 
91.7 
91.0 
90.3 
89.6 
88.9 

111.2 
110.4 
109.6 
108.8 
108.0 
in:  .-J 
106.4 

ior..6 

104.8 
104.0 
103.2 
102.4 
101.6 

""                  159 

TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  340  L.  531.] 

[No.  379  L.  579. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

340 
1 
2 
3 
4 
5 
6 

7 
8 
9 

350 
1 
2 
3 

4 

5 
6 

r» 

8 
9 

360 
1 
2 
3 

4 
5 

6 
7 
8 
9 

370 

1 

2 
3 
4 
5 
6 
7 
8 
9 

531479 
2754 
4026 
5294 
6558 
7819 
9076 

1607 
2882 
4153 
5421 
6685 
7945 
9202 

1734 

3009 
4280 
5547 
6811 
8071 
9327 

1862 
3136 
4407 
5674 
6937 
8197 
9452 

1990  | 
3264 
4534 
5800 
7063 
8322 
9578 

2117 
j  3391 
I  4661 
5927 
7189 
8448 
9703 

2245 
3518 
4787 
6053 
7315 
8574 
9829 

2372 
3045 
4914 
6180 
7441 
8699 
9954 

1205 
2452 
309G 

4936 
61?'2 
7405 
8035 
9861 

2500 
3772 
5041 
6306 
7567 
8825 

2627 
3899 
5167 
6432 
7693 
8951 

128 
127 
127 
126 
126 
126 

125 
125 
125 
124 

124 
124 

123 
123 

123 
123 

122 
121 
121 
121 

120 
120 
120 

119 
119 
119 
119 
118 
118 
118 

117 

J17 
117 
116 
116 
116 
115 
115 
115 
114 

0079 
1330 
2576 
3820 

5060 
6290 
7529 
8758 
9984 

0204 
1454 
2701 
3944 

5183 
6419 
7052 
8881 

540329 
1579 
2825 

4068 
5307 
6543 

7775 
9003 

0455 
1704 
2950 

4192 
5431 
6666 
7898 
9126 

0580 
1829 
3074 

4316 
6555 
6789 
8021 
9249 

0705 
1953 
3199 

4440 
5678 
6913 
8144 
9371 

0830 
2078 
3323 

4564 

5802 
7036 
8267 
9494 

0955 
2203 
3447 

4688 
5925 
7159 
8389 
9616 

1080 
2327 
3571 

4812 
6049 
7282 
8512 
9739 

0106 
1328 
2547 
3702 
4973 
6182 

7387 
8589 
9787 

550228 
1450 
2668 
3883 
5094 

6303 
7507 
8709 
9907 

0351 
1572 
2790 
4004 
5215 

6423 
7627 

8829 

0473 
1694 
2911 
4126 
5336 

6544 

7748 
8948 

0595 
1816 
3033 
4247 
5457 

6664 

7868 
90(58 

0717 
1938 
3155 
4368 
5578 

6785 

7988 
9188 

0840 
2000 
3276 
4489 
5699 

6905 
8108 
9308 

0962 
2181 
3398 
4610 
5820 

7026 
8228 
9428 

1084 
2303 
3519 
4731 
5940 

7146 
8349 
9548 

1206 
2425 
3040 
4852 
6001 

7267 
8469 
9007 

0026 
1221 

2412 
3600 
4784 
5966 
7144 

8319 
9491 

0146 
1340 
2531 
3718 
4903 
6084 
7262 

8436 
9608 

0265 
1459 
2650 
3837 
5021 
6202 
7379 

8554 
9725 

0385 
1578 
2769 
3055 
5139 
6320 
7497 

8671 

9842 

!  0504 
1098 
2887 
4074 
5257 
6437 
7614 

8788 
9959 

0624 
1817 
3006 
4192 
5376 
6555 
7732 

8905 

0743 
1936 
3125 
4311 
5494 
6673 
7849 

9023 

0863 
2055 
3244 
4429 
5612 
6791 
7967 

9140 

0982 
2174 
3362 
4548 
5730 
6909 
8084 

9257 

561101 
2293 
3481 
4666 
5848 
7026 

8202 
9374 

0076 
1243 

2407 
3568 
4726 
5880 
7032 
8181 
9326 

0193 
1359 
2523 
3684 
4841 
5996 
7147 
8295 
9441 

0309 
1476 
2639 
3800 
4957 
6111 
7262 
8410 
9555 

0426 
1592 
2755 
3915 
5072 
6226 
7377 
8525 
9069 

570543 
1709 
2872 
4031 
5188 
6341 
7492 
8639 

0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 

0776 
1942 
3104 
4263 
5419 
6572 
7722 
8868 

0893 
2058 
3220 
4379 
5534 
6687 
7836 
8983 

1010 
2174 
3336 
4494 
5650 
6802 
7951 
9097 

1126 
2291 
3452 
4610 
5765 
6917 
8066 
9212 

PROPORTIONAL  PARTS. 

Diff.   1 

2      3 

4 

5 

6 

7 

8 

9 

128   12.8 
127   12  7 
126   12  6 
125   12.5 
124   12.4 
123   12.3 
122   12.2 
121   12.1 
120   12.0 
119   11.9 

25.6    38.4 
25  4    38.1 
25.2    37.8 
25.0    37.5 
24.8    37.2 
24.6    36.9 
24.4    36.6 
24.2    36.3 
24,0    36,0 
23.8    35.7 

51.2 
50.8 
50.4 
50.0 
49.6 
49.2 
48.8 
48.4 
48.0 
47.6 

64.0 
63.5 
63.0 
62.5 
62.0 
61.5 
61.0 
60.5 
60.0 
59.5 

76.8 
76.2 
75.6 
75.0 
74.4 
73.8 
73.2 
72.6 
72.0 
71.4 

89.6 

88.9 
88.2 
87.5 
86  8 
86.1 
85.4 
84.7 
84.0 
83.3 

102.4 
101.6 
100.8 
100.0 
99.2 
98.4 
97.6 
96.8 
96.0 
95.2 

115.2 
114.3 
113.4 
112.5 
111.6 
110.7 
109.8 
108.9 
108.0 
107.1 

160 


TABLE   XI.— LOGARITHMS   OF    NUMBERS. 


No.  380.  L.  579.] 

[No.  414  L.  617. 

N. 

0 

1 

2 

3 

* 

5 

6 

7 

8 

9 

Diff. 

380 

579784 

9898 

0012 

01 

26 

0241 

0355 

0469 

0583 

069 

7 

0811 

1 
2 

580925 
2063 

1039 
2177 

1153 
2291 

1267 
2404 

1381 
2518 

1495 
2631 

1608 
2745 

1722 
2858 

1836 
2972 

1950 
3085 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

4 

4331 

4444 

4557 

46 

70 

4783 

4896 

5009 

5122 

523 

5 

5348 

113 

5 

5461 

5574 

5686 

57 

99 

5912 

6024 

6137 

6250 

636 

I 

6475 

6 

6587 

6700 

6812 

69 

25 

7037 

7149 

7262 

7374 

7486 

7599 

7 

7711 

7823 

7935 

80 

47 

8160 

8272 

8384 

8496 

860 

s 

8720 

112 

8 
9 

8832 
9950 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

0061 

0173 

0284 

0396 

0507 

0619 

0730 

0842 

0953 

390 

591065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

1 

2177 

2288 

2399 

25 

10 

2621 

2732 

2843 

2954 

306 

4 

3175 

111 

2 

3286 

3397 

3508 

36 

18 

3729 

3840 

3950 

4061 

41? 

1 

4282 

3 

4393 

4503 

4614 

47 

24 

4834 

4945 

5055 

5165 

52? 

li 

5386 

4 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

5 

6597 

6707 

6817 

69 

27 

7037 

7146 

7256 

7366 

74? 

ti 

7586 

110 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

7 

8791 

8900 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

8 

9883 

9992 

0101 

0210 

0319 

0428  !  0537 

0646 

0755 

0864 

109 

9 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

400 

2060 

2169 

2277 

23 

36 

2494 

2603 

2711 

2819 

2928 

3036 

1 

3144 

3253 

3361 

34 

EJ9 

3577 

3686 

3794 

3902 

4010 

4118 

108 

2 

422(5 

4334 

4442 

45 

~>o 

4658 

4766 

4874 

4982 

508 

9 

5197 

3 

5305 

5413 

5521 

56 

28 

5736 

5844 

5951 

6059 

610 

6 

6274 

4 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

5 

7455 

7562 

7669 

77 

7884 

7991 

8098 

8205 

831 

0 

8419 

107 

6 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

9594 

9701 

9808 

99 

14 

0021 

0128 

0234 

0341 

044 

7 

0554 

8 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

105 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

1 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

2 

4897 

5003 

5108 

52 

13 

5319 

5424 

5529 

5634 

574 

0 

5845 

3 

5950 

6055 

6160 

62 

i5 

6370 

6476 

6581 

6686 

679 

0 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3 

4 

5 

6     7 

8 

9 

118 

11.8 

23.6 

35.4 

47.2 

59.0 

70.8    82.6 

94.4 

106.2 

117 

11.7 

23.4 

35.1 

46.8 

58.5 

70.2    81.9 

93.6 

105.3 

116 

11.6 

23.2 

34.8 

46.4 

58.0 

69.6    81.2 

92.8 

ldl.4 

115 

11.5 

23.0 

34.5 

46.0 

57.5 

69.0    80.5 

92.0 

103.5 

114 

11.4 

22.8 

34.2 

45.6 

57.0 

68.4    79.8 

91.2 

1<>'2  (5 

113 

11.3 

22.6 

.33.9 

45.2 

56.5 

67.8    79.1 

90.4 

101.7 

112 

11.2 

22.4 

33.6 

44.8 

56.0 

67.2    78.4 

89.6 

100.8 

111 

11.1 

22.2 

33.3 

44.4 

55.5 

66.6    77.7 

88.8 

99.9 

110 

11.0 

22^0 

33.0 

44.0 

55.0 

66.0    77.0 

88.0 

99.0 

109 

10.9 

21.8 

32.7 

43.6 

54.5 

65.4    76.3 

08.1 

108 

10.8 

21.6 

32.4 

43.2 

54.0 

64.8    75.6 

86.4 

97.2 

107 

10.7 

21.4 

32.1 

42.8 

53.5 

64.2    74.9 

85.6 

96.3 

106 

10.6 

21  2 

31.8 

42.4 

53.0 

63.6    74.2 

84.8 

06.4 

105 
105 

10.5 
10.5 

2l!5 
21.0 

31.5 
31.5 

42.0 
42.0 

52.5 
52.5 

63.0    73.5 
63.0    73.5 

84.0 
84.0 

94.5 
94.5 

104 

10.4 

20.8 

31.2 

41.6 

52.0 

62.4    72.8 

83.2 

93.6 

161 


TABLE    XI. — LOGARITHMS    OF    LUMBERS. 


No.  415  L.  618.]   '                                [No.  459  L.  662 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

415 

618048 

8153 

8257 

8362 

8466 

8571 

8676  8780  8884 

8989 

105 

6 

9093 

9198 

9302 

9406 

9511 

9615 

9719  9824 

9928 

rjAOO 

- 

620136 

0240 

0344 

0448 

0552  |  0656 

0760 

0864  0968 

UlMM 

1072 

104 

8 

1176 

1280 

1384 

1488 

1592   1695 

1799 

1903  2007 

2110 

9 

2214 

2318 

2421 

2525 

2628   2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

1 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004  I  5107 

5210 

103 

2 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

3 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7203 

4 

7366 

7468 

7571 

7673 

7775 

787'8 

7980 

8082 

8185 

8287 

5 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

9410 

9512 

9613 

9715 

9817 

9919 

0021 

0123 

0224 

0326 

y 

630428 

0530 

0631 

0733 

0835 

0936 

•  1038 

1139 

1241 

1342 

8 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

9 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

101 

1 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

2 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287  6388 

3 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290  7390 

4 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290  8389 

100 

5 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287  9387 

6 

9486 

9586 

9686 

9785 

9885 

9984 





'  

0084 

0183 

0283 

0382 

7 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

9 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

1 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

5324 

2 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

3 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

4 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

5 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

6 

9335 

9432 

9530 

9627 

9724 

9821 

9919 







0016 

0113 

0210 

7 

650308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

8 

1278 

1375 

14V2 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

9 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

450 

3213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

1 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

4850 

.4946 

5042 

2 

5138 

5235 

5331 

5427 

5523 

5019 

5715 

5810 

5906 

6002 

96 

3 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

4 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

5 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

6 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726  |  9821 

7 

9916 

I 

0011 

0106 

0201 

0296 

0391 

0486 

0581 

0676  0771 

95 

8 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623   1718 

9 

1813 

1907 

2002 

2096 

2191 

2286 

2380 

2475 

2569  2603 

PROPORTIONAL  PARTS. 

Diff   1 

234 

5 

678 

9 

105   10  5 

21.0    31  5    42.0 

52  5 

63  0    73.5    84  0 

94,5 

104   lo  4 

20  8    31  2    41  6 

52.0 

62  4    72  8    83  2 

93.6 

103   10  3 

206    309    41.2 

51  5 

61  8    72  1    82.4 

92  7 

102   10  2 

20  4    30  6  !  40  8 

51  0 

61  2    71.4    81  6 

91  8 

101   10  1 

20  2    30  3    40.4 

50  5 

60  6    70  7    80  8 

90.9 

100   10.0 

20  0    30.0    40  0 

50  0 

60.0    70  0    80  0 

90  0 

99    99 

19  8    29  7    39  6 

49  5 

59  4    69  3    79  2 

89.1 

162 


TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  460  L.  662.] 

[No.  499  L.  698. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

460 
1 
o 

3 
4 
5 
6 

8 
9 

470 
1 
2 
3 
4 
5 
6 
7 
8 

9 

480 
1 
2 
3 
4 
5 
6 
7 
8 
9 

490 
1 
2 
3 
4 
5 
6 
7 
8 
9 

662758 
3701 
4642 
5581 
6518 
7453 
8386 
9317 

2852 
3795 
4736 
5675 
6612 
7546 
8479 
9410 

2947 
3889 
4830 
5769 
6705 
7640 
8572 
9503 

3041 
3983 
4924 

7733 

8665 
9596 

3135 

4078 
5018 
5956 
6892 
7826 
8759 
9689 

3230 
4172 
5112 
6050 
6986 
7920 
8852 
9782 

3324 

5206 
6143 
7079 
8013 
8945 
9875 

3418 
4300 
5299 

7173 
8106 
9038 
9967 

3512 

1  15  1 

5898 

6331 
7260 
8199 
9131 

3607 
4548 
5487 
6424 
7360 
8293 
9224 

94 

93 

92 
91 

90 
89 

88 
87 

0060 
0988 
1913 

2836 
3758 
4677 
5595 
6511 
7424 
8336 
9246 

0153 
1080 
2005 

2929 
3850 
4769 
5687 
6602 
7516 
8427 
9337 

670246 
1173 

2098 
3021 
3942 

4861 
5778 
6094 
7607 
8518 

0339 
1265 

2190 
3113 
4034 
4953 
5870 
6785 
7698 
8609 
9519 

0431 
1358 

2283 
3205 
4126 
5045 
5962 
6876 
7789 
8700 
9610 

0524 
1451 

2375 
3297 
4218 
5137 
6053 
6968 

8791 
9700 

0617 
1543 

2467 
3390 
4310 
5228 
6145 
7059 
7972 
8882 
9791 

0710 
1636 

2560 
3482 
4402 
5320 
6236 
7151 
8063 
8973 
9882 

0802 
1728 

2652 
3574 
4494 
5412 
6328 
7242 
8154 
9064 
9973 

0895 
1821 

2744 
3666 
4586 
5503 
6419 
7333 
8245 
9155 

0063" 
0970 

1874 
2777 
3677 
4576 
5473 
6368 

8153 
9042 

0154 
1060 

1964 
3867 
3767 
4666 
5563 
6458 
7351 
8242 
9131 

0245 
1151 

2055 
2957 
3857 
4756 
5652 
6547 
7440 
8331 
9220 

680336 

1241 
2145 
3047 
3947 
4845 
5742 
6636 
7529 
8420 
9309 

690196 
1081 
1965 
2847 
3727 
4605 
5482 
6356 
7229 
8100 

0426 

1332 
2235 
3137 
4037 
4935 
5831 
6726 
7618 
8509 
9398 

0517 

1422 
2326 
3227 
4127 
5025 
5921 
6815 
7707 
8598 

0607 

1513 
2416 

asi7 

4217 
5114 
6010 
6904 
7796 
8687 
9575 

0698 

1603 
2506 
3407 
4307 
5204 
6100 
6994 
7886 
8776 
9664 

0550 
1435 
2318 
3199 
4078 
4956 
5832 
6706 
7578 
8449 

0789 

1693 
2596 
3497 
4396 
5294 
6189 
7083 
7975 
8865 

0879 

1784 
2686 
3587 
4486 
5383 
6279 
7172 
8064 
8953 

0019 

0905 
1789 
2671 
3551 
4430 
5307 
6182 
7055 
7926 
8796 

0107 

0993 
1877 
2759 
3639 
4517 
5394 
6269 
7142 
8014 
8883 

0285 
1170 
2053 
2935 
3815 
4693 
5569 
6444 
7317 
8188 

0373 
1258 
2142 
3023 
3903 
4781 
5657 
6531 
7404 
8275 

0462 
1347 
2230 
3111 
3991 
4868 
5744 
6618 
7491 
8362 

0639 
1524 
2406 
3287 
4166 
5044 
5919 
6793 
7665 
8535 

0728 
1612 
2494 
3375 
4254 
5131 
6007 
6880 
7752 
8622 

0816 
1700 
2583 
3463 
4342 

6094 
6968 
7839 
8709 

PROPORTIONAL  PARTS. 

Diff 

1 

2      3 

4 

5 

6 

7 

8 

9 

98 
97 
96 
95 
94 
93 
92 
91 
90 
89 
88 
87 
86 

9.8 
9.7 
9.6 
9.5 
9.4 
9.3 
9.2 
9.1 
9.0 
8.9 
8.8 
8.7 
8.6 

19.6    29.4 
19.4    29.1 
19.2    28.8 
19.0    28.5 
18.8    28.2 
18.6    27.9 
18.4    27.6 
18.2    27.3 
18.0    27.0 
17.8    26.7 
17.6    26.4 
17.4    26.1 
17.2    25.8 

39.2 
38.8 
38.4 
38.0 
37.6 
37.2 
36.8 
36.4 
36.0 
35.6 
35.2 
34.8 
34.4 

49.0 
48.5 
48.0 
47.5 
47.0 
46.5 
46.0 
45.5 
45.0 
44.5 

ftg 

43.0 

58.8 
58.2 
57.6 
57.0 
56.4 
55.8 
55.2 
54.6 
54.0 
58.4 
52.8 
'52.2 
51.6 

68.6 
67.9 
67;2 

65.'8 
65.1 
64.4 
63.7 
63.0 
62.3 
61.6 
60.9 
60.2 

78.4 
77.6 
76.8 
76.0 
75.2 
74.4 
73.6 
72.8 
72.0 
71.2 
70.4 
69.6 
68.8 

88.2 
87.3 
86.4 
85.5 
84.6 
83.7 
82.8 
81.9 
81.0 
80.1 
79.2 

n 

163 

TABLE   XI. — LOGARITHMS   OF   NUMBERS. 


No.  500  L.  698.]                                  [No.  544  L.  736. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

1 

9838 

9924 

0011 

0098 

0184 

0271 

0358 

0444 

0531 

0617 

2 

700704 

0790 

0877 

0963 

1050 

113G 

1222 

1309 

1395 

1482 

3 

1568 

1654 

1741 

1827 

1913 

1990 

2085 

2172 

2258 

2344 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

8 

5864 

5949 

6035 

6120 

62G6 

6291 

6376 

6462 

6547 

6632 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

510    7570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

85 

2 

9270 

9355 

9440 

9524 

9609 

9694. 

9779 

9863 

9948 

0033 

3 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

4 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

5 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3407 

7 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

1 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

83 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

0077 

5 

720159 

0242 

0325 

0407 

0490 

0573 

0655 

0738 

0821 

0903 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

4276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

49bl 

5013 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

2 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

6 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

7 

9974 

0055 

0136 

0217 

0298 

0378 

0459 

0540 

0621 

0702 

8 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

540 

2394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

1 

3197 

3278 

3358 

3438 

3518 

8598 

3679 

3759 

3839 

3919 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

Qf\ 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5519 

oU 

4 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3      4 

5 

678 

9 

87 

8.7 

17.4 

26  1    34.8 

43  5 

52  2    60.9    69  6 

78  3 

86 

8.6 

17.2 

25.8    34.4 

43  0 

51  6    60  2    68.8 

77  4 

85 

8.5 

17.0 

25  5    34.0 

42  5 

51.0    59.5    68.0 

76  5 

84 

8.4 

16.8 

25.2    33.6 

420 

50.4    58  8    67.2 

75.6 

164 


TABLE  XI.— LOGARITHMS   OP  NUMBERS. 


No.  545  L.  736.] 

|No.  584  L.  707. 

N. 

0 

1 

2 

8 

4  ||  6 

C 

7 

8 

9   Diff. 

545 

7 
8 
9 

736397 
7193 
7987 
8781 
9572 

6476 
7272 

8067 
8860 
9651 

6556 
7352 
8146 
8939 
9731 

6635 
7431 
8225 
9018 
9810 

6715 
7511 
8806 

9097 
9889 

,  6795 
1  7590 
i  8384 
91  77 
9968 

6874 
7670 
8463 
9256 

6954 
7749 
8543 
9335 

7034  7113 

7829  7908 
8622  8701 
9414   9493 

550 
1 
2 
3 

4 
5 
6 
7 
8 
9 

740363 
1152 
1939 
2725 
3510 
4293 
5075 
5855 
6634 
7412 

0442 
1230 
2018 
2804 
3588 
4371 
5153 
5933 
6712 
7489 

0521 
1309 
2096 
2882 
3667 
4449 
5231 
6011 
6790 
7567 

0600 
1388 
2175 
2961 
3745 
4528 
5309 
6089 
6868 
7645 

0678 
1467 
2254 
3039 
3823 
4606 
5387 
6167 
6945 
7722 

0757 
!  1546 
,  2332 
3118 
3902 
4684 
5465 
6245 
7023 
7800 

0047 

0836 
1624 
2411 
3196 
3980 
4762 
5543 
6323 
7101 
7878 

0126 

0915 
1703 
2489 
3275 
4058 
4840 
5621 
6401 
7179 
7955 

0205 

0994 
1782 
2568 
3353 
4136 
4919 
5699 
6479 
7256 
8033 

0284 

1073 
1860 
2647 
3431 
4215 
4997 
5777 
6556 
7334 
8110 

79 

78 

560 
1 

8188 
8963 

8266 
9040 

8343 
91J8 

8421 
9195 

8498 
9272 

8576  8653 
9350  9427 

8731 
9504 

8808 
QfiflB 

8885 
9659 

2 

9736 

9814 

9891 

qc 

96 

3 

4 
5 
6 

7 
8 

0045 
0817 
1587 
2356 
3123 
3889 
4654 

0123  0200 
0894  0971 
!  16U4  1741 
|  2433  2509 
3200  3277 
3966  4042 
4730  4807 

0277 
1048 
1818 
2586 
3353 
4119 
4883 

0354 
1125 
1895 
2663 
3430 
4195 
4960 

0431 
1202 
1972 
2740 
3506 
4272 
5036 

77 

750508 
1279 
2048 
2816 
3583 
4348 

0586 
1356 
2125 
2893 
3660 
4425 

0663 
1433 
2202 
2970 
3736 
4501 

0740 
1510 
2279 
3047 
3813 
4578 

9 

5112 

5189 

5265  |  5341 

5417 

5494  5570 

5646 

5722 

5799 

570 

5875 

5951 

6027 

6103 

6180 

6256  !  6332 

6408 

6484 

6560 

1 

6636 

6712 

6788 

68 

>4 

6940 

7016  7093 

7168 

734 

4 

7320 

76 

2 

7396 

7472 

7548 

7(5 

24 

7700 

7775  7851 

7927 

800 

R 

8079 

3 

8155 

8230 

8306 

83 

*2 

8458 

8533  8609 

8685 

8761 

8836 

4 

8912 

8988 

9063 

91 

J,9 

9214 

i  9290  9366 

9441 

951 

1» 

9592 

5 

9668 

9743 

9819 

06 

9-1 

9970 

n/v<pr  ,  nioi 

1!  1  III' 

6 

760422 

0498 

0573 

0649 

0724 

0799  0875 

0950 

1025 

1101 

7 

1176 

1251 

1326 

14 

)2 

1477 

i  1552  1627 

1702 

177 

8 

1853 

8 

1928 

2003 

2078 

21 

53 

2228 

2303  2378 

2453 

252 

D 

2604 

9 

2679 

2754 

2829 

2904 

2978 

i  3053  j  3128 

3203 

3278 

3353 

73 

580 

3428 

3503 

&578 

36 

>3 

3727 

!  3802 

3877 

3952 

4027 

4101 

1 

4176 

4251 

4326 

4400 

4475 

4550  i  4624 

4699 

477 

4 

4848 

4923 

4998 

5072 

51 

47 

5221 

5296  j  5370 

5445 

552 

0 

5594 

3 

5669 

5743 

5818 

58 

12 

5966 

6041 

6115 

6190 

626 

4 

6338 

4 

6413 

6487 

6562 

66 

36 

6710 

6785 

6859 

6933 

7007 

7082 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

83 

8.3 

16.6 

24.9 

33.2 

41.5 

49.8 

58.1 

66.4 

74.7 

82 

8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4 

65.6 

73.8 

81 

8.1 

16.2 

24.3 

32.4 

40.5 

48.6 

56.7 

64.8 

72.9 

80 

8.0 

16.0 

24.0 

32.0 

40.0 

48.0 

56.0 

64.0 

72.0 

79 

7.9 

15.8 

23.7 

31.6 

39.5 

47.4 

55.3 

63.2 

71.1 

78 

7.8 

15.6 

23.4 

31.2 

39.0 

46.8 

54.6 

62.4 

70.2 

77    7.7 

15.4 

23.1 

30.8 

38.5 

46.2 

53.9 

61.6 

69.3 

76 

7  6 

15.2 

22.8 

30.4 

38.0 

45.6 

53.2 

60.8 

68.4 

75 

7.5 

15.0 

22.5 

30.0 

37.5    45.0 

52.5 

60.0 

67.5 

74 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.8 

59.2 

66.6 

165 


TABLE   XI. — LOGAKITHMS    OF    NUMBERS. 


No.  585  L.  767.1                                  [No.  629  L.  799. 

N. 

0 

1 

2 

3 

4 

5  |  6 

7 

8 

9 

Diff. 

585 

767156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

6 

7898 

7972 

8046 

8120 

8194 

i  8268 

8342 

8416 

8490 

8564 

74 

7 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

g 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

0042 

9 

770115 

0189 

02C3 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

8 

6701 

6774 

6S46 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

9 

7427 

7499 

7572 

7644 

7717 

7789 

.7862 

7934 

8006 

8079 

600 

8151 

8224 

8296 

8368 

8441 

8513 

8585 

8658 

8730 

8802 

1 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

2 

9596 

9669 

9741 

9813 

9885 

9957 

0029 

0101 

0173 

0245 

3 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

6 

2473 

2544 

2616 

2688 

2T59 

2831 

2902 

2974 

3046 

3117 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

9581 

9651 

9722 

9792 

9863 

9933 

0004 

0074 

0144 

QOIPJ 

7 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1*50 

1620 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

3 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

6 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7i29 

7198 

7 

7268 

7337 

7406 

7475 

7545 

!  7614 

7683 

7752 

7821 

7890 

8 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

9 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

75    7.5 

15.0    22.5    30.0 

37.5 

45.0    52.5    60.0 

67.5 

74    7.4 

14.8    22.2    29.6 

37.0 

44.4    51.8    59.2 

66.6 

73    7.3 

14.6    21.9    29.2 

36.5 

43.8    51.1    58.4 

65.7 

72    7.2 

14.4    21.6    28.8 

36.0 

43.2    50.4    57.6 

64.8 

71    7.1 

14.2    21.3    28.4 

35.5 

42.6    49.7    56.8 

63.9 

70    7.0 

14.0    21.0    28.0 

35.0 

42.0    49.0    56.0 

63.0 

69    6.9 

13.8    20.7    27.6 

34.5 

41.4    48.3    55.2 

62.1 

166 


TABLE    XL— LOGARITHMS    OF    NUMUEUS. 


No.  630  L.  799.] 

[No.  674  L.  829. 

.-N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diflf. 

'630 

799341 

9409 

9478 

9547 

9616 

9685 

9754 

9323 

9892 

9961 

1 
2 

800029 
0717 

0098 

0786 

0167 
0854 

0236 
0923 

0305 
0992 

0373 
1061 

0442 
1129 

0511 
1198 

0580 

0648 
1335 

3 

1404 

1472 

1541 

16 

J9 

1678 

1747 

1815 

1884 

1062 

2021 

4 

2089 

2158 

2226 

22 

J5 

2363 

2432 

2500 

2 

508 

281 

7 

2705 

5 

2774 

2842 

2910 

29 

re 

3047 

3116 

3184 

3 

>:>i> 

332 

i 

3389 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

7 

4139 

4208 

4276 

4& 

14 

4412 

4480 

4548 

4 

616 

468- 

r> 

4753 

8 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

640 

806180 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

6790 

1 

6858 

6926 

6994 

70 

31 

7129 

7197 

7264 

332 

740 

0 

7467 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

3 

8211 

8279 

8346 

84 

14 

8481 

8549 

8616 

8 

684 

875 

i 

8818 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

5 

9560 

9627 

9694 

97 

W 

9829 

9896 

9964 

(\ 

no-i 

A/yi 

o 

6 

810233 

0300 

0367 

0434 

0501 

0569 

0636 

UVOA 

0703 

UU»O 

0770 

0165 
0837 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

17 

~6 

1843 

1910 

1977 

2 

044 

211 

1 

2178 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713- 

2780 

2847 

650 

2913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

1 

3581 

3648 

3714 

37 

SI 

3848 

3914 

3981 

4 

048 

411 

4 

4181 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

3 

4913 

4980 

5046 

51 

13 

5179 

5246 

5312 

5 

378 

544 

5 

5511 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

5 

6241 

6308 

6374 

64 

40 

6506 

6573 

6639 

6 

705 

677 

1 

6838 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

7 

7565 

7631 

7698 

77 

54 

7830 

7896 

7962 

8 

02S 

SOS 

4 

8160 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

aa 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

DO 

660 

9544 

9610 

9676 

97 

11 

9807 

9873 

9939 

0 

nai 

007 

n 

0136 

1 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

2 

0858 

0924 

0989 

10 

>5 

1120 

1186 

1251 

1 

317 

138 

2 

1448 

3 

1514 

1579 

1645 

17 

10 

1775 

1841 

1906 

1 

972 

203 

M 

2103 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

6 

3474 

3539 

3605 

36 

70 

3735 

3800 

3865 

3 

930 

39S 

a 

4061 

7 

4126 

4191 

4256 

43, 

21 

4386 

4451 

4516 

4 

581 

484 

(5 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

528 

0 

5361 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

659 

8 

6658 

1 

6723 

6787 

6852 

69 

7 

6981 

7046 

7111 

7 

175 

724 

0 

7:305 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

3 

8015 

8080 

8144 

82( 

)',) 

8273 

8338 

8402 

8 

467 

853 

i 

8595 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

PROPORTIONAL  PARTS. 

.'Biff   1 

2 

3 

4 

5 

6 

7 

8 

9 

68    68 

13  6 

20  4 

27  2 

34  0 

40  8 

47  6 

54  4 

61  3 

67    67 

13  4 

20.1 

26  8 

33  5 

40  2 

4li  !> 

IK)  -i 

66    66 

13.2 

19  8 

26:4 

33  0 

39  0 

41  i  2 

52  8 

r>:>  4 

65    65 

13  0 

19  5 

26  0 

32.5 

39  0 

i:>  :> 

58  5 

64    6.4 

12  8 

19.2 

25  6 

32.0 

;;s.  i 

il  N 

51  2    ov.o 

167 

TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  675  L.  829.]                                 [No.  719  L.  857. 

N. 

0 

1 

2 

I 

4 

6 

6 

7 

8 

9 

Diff. 

675 
g 

829304 
9947 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

0011 

0075 

0139 

0204 

0268 

0332 

0396 

0460 

0525 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

8 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

680 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

3 

4421 

4484 

4548 

4611 

4675  :  4739 

4802 

4866 

4929 

4993 

4 

5056 

5120 

5183 

5247 

6310 

5373 

5437 

5500 

5564 

5627 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

1 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 



0043 

2 

840106 

0169 

0232 

0294 

0357 

0420 

0482 

0545 

0608 

0671 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

700 

5098 

5160 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

G090 

6151 

6213 

6275 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235  i  9297 

9358 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849  |  9911 

9972 

8 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462  i  0524  0585 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

710 

1258 

132.0 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968  3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577  3637 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185  4245 

5 

4306 

4367 

4428 

4488 

4549  . 

4610 

4670 

4731 

4792  I  4852 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

7 

5519 

5580 

5640 

5701 

5761  i 

5822 

5882 

5943 

6003 

6064 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

65    6.5 

13.0    19.5    26.0 

32.5 

39.0    45.5    52.0 

58.5 

64    6.4 

12.8    19.2    25.6 

32.0 

38.4    44.8    51.2 

57.6 

63    6.3 

12.6    18.9    25.2 

31,5 

37.8    44.1    50.4 

56.7 

62    6.2 

12.4    18.6    24.8 

31.0 

37.2    43.4    49.6 

55.8 

61    6.1 

12.2    18.3    24.4 

30.5 

36.6    42.7    48.8 

54.9 

60    6.0 

12.0    18.0    24.0 

30.0 

36.0    42.0    48.0 

54.0 

168 


TABLE    XI. — LOGARITHMS  OF   NUMBERS. 


No.  720  L.  857.]                                 [No.  764  L.  883. 

N. 

0 

1 

2 

8 

4 

6 

e 

7 

8 

9 

Diff. 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

3 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

9799 

9859 

9918 

9978 

0038 

(  NI'IS! 

m  f^ft 

5 

860338 

0398 

0458 

0518 

0578 

0637 

wyo 
0697 

UltX) 

0757 

0218 
0817 

0278 
0877 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2439 

2549 

2608 

2668 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3114 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

5 

6287 

6346 

6405 

6465 

6524 

6583 

C642 

6701 

6760 

6819 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

91.3 

740 

9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

1 

9818 

9877 

9935 

9994 

0053 

0111 

0170 

0228 

0287 

0345 

2 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2008 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

8146 

3204 

3262 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

|  4772 

4830 

4888 

4945 

5003 

750 

5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

6 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

7 
g 

9096 
9669 

9153 
9726 

9211 

9784 

9268 
9841 

9325 

9898 

9383 
9956 

9440 

9497 

9555 

9612 

0013 

0070 

0127 

0185 

9 

880242 

0299 

0356 

0413 

0-171 

0528 

0585 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

ms 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1SJJS 

K7 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

Oi 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3      4 

5 

6      7.     8 

9 

59 

5.9 

11.8 

17.7    23.6 

29  5 

35.4    41.3    47.2 

53.1 

58 

5.8 

11.6 

17.4    23.2 

29^0 

34.8    40.6    46.4 

52.2 

57 

5.7 

11.4 

17.1    22.8 

28.5 

34.2    39.9    45.6 

51.3 

56 

5.6 

11.2 

16.8    22.4 

28.0 

33.6    39.2    44.8 

50.4 

169 


TABLE   XI. — LOGARITHMS    OF    NUMBERS. 


No.  765  L.  883.]                                  [No.  809  L.  908. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

765 

883661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

7 

4?95 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

5 
6 

9302 
9862 

9358 
9918 

9414 
9974 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

0030 

0086 

0141 

0197 

0253 

0309 

0365 

7 

890421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

1 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

2 

8725 

8780 

8&S5 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

3 
4 

92Y3 

9821 

9328 
9875 

9383 
9930 

9437 
9985 

9492 

9547 

9602 

9656 

9711 

9766 

0039 

0094 

0149 

0203 

0258 

0312 

5 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

6 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

800 

3090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

•4066 

4120 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

77'87 

7841 

7895 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

6      7 

8 

9 

57    5.7 

11.4    17.1    22.8 

28.5 

34.2    39.9 

45.6 

51.3 

56    5.6 

11.2    16.8    22.4 

28.0 

33.6    39.2 

44.8 

50.4 

55    5.5 

11.0    16.5    22.0 

27.5 

33.0    38.5 

44.0 

49.5 

54    5.4 

10.8    16.2    21.6 

27.0  |  32.4    37.8 

43.2 

48.6 

170 


TABLE    XI. — LOGARITHMS   OF    NUMBERS. 


No.  810  L.  908.]                                  [No.  a>i~L~9^T 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

810 
1 
2 

908485 
9021 
9556 

8539 
9074 
9610 

8592 
9128 
9663 

8646 
9181 
9716 

8699 
9235 
9770 

8753 
9289 
9823 

8807 
9342 
9877 

8860 
9396 
9930 

8914 
9449 
9984 

8967 
9503 

3 
4 
5 
6 

8 
9 

910091 
0624 
1158 
1690 
2222 
2753 
3284 

0144 
0678 
1211 
1743 
2275 
2806 
3337 

0197 
0731 
1264 
1797 
2328 
2859 
3390 

0251 
0784 
1317 
1850 
2381 
2913 
3443 

0304 
0838 
1371 
1903 
2435 
2966 
3496 

1  0358 
0891 
1424 
1956 
2488 
3U19 
3549 

0411 
0944 
1477 
2009 
2541 
3072 
3602 

0464 
0998 
1530 
2063 
2594 
3125' 
3655 

0518 
1051 
1584 
2116 
2647 
3178 
3708 

0037 
0571 
1104 
1G37 
2169 
2700 
3231 
3761 

53 

820 
1 
2 

3814 
4343 

4872 

3867 
4396 
4925 

3920 
4449 
4977 

3973 
4502 
5030 

4026 
4555 
5083 

4079 
4608 
5136 

4132 
4660 
5189 

4184 
4713 
5241 

4237 
4766 
5294 

4290 
4819 
5347 

3 
4 
5 
6 

5400 
5927 
6454 

6980 

5453 

5980 
6507 
7033 

5505 
6033 
6559 

7085 

5558 
6085 
6612 
7138 

5611 
6138 
6664 
7190 

5664 
6191 
6717 
7243 

5716 
6243 
6770 

7295 

5769 
6296 
1822 
7348 

5822 
6349 
6875 
7400 

5875 
6401 
6927 
7453 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397  8450 

8502 

9 

8555 

8607 

8659 

8712 

87&4 

8816 

8869 

8921- 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

1 

9601 

9653 

9706 

9758 

9810 

9802 

9914 

9907 

2 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

UU1\» 

0541 

OOil 
0593 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

4 

1166 

1218 

127'0 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

SS57 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9888 

850 

9419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

1  1   9930 

9981 

51 

0032 

0083 

0134 

0185 

0236 

0287 

0338 

0389 

2 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

3      4 

5 

678 

9 

53    5.3 

10.6 

15.9    21.2 

26.5 

31.8    37.1    42.4 

47.7 

52    5.2 

10.4 

15.6    20.8 

26.0 

31.2    36.4    41.6 

46.8 

51    5.1 

10.2 

15.3    20.4 

25.5 

30.6    35.7    40.8 

45.9 

50    5.0 

10.0 

15.0    20.0 

25.0 

30.0    35.0    40.0 

45.0 

171 


TABLE    XI. — LOGARITHMS    OF    NUMBERS. 


No.  855  L.  931.1                                  [No.  899  L.  954. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

855 

931966 

2017 

2068 

2118 

2169 

2220 

»r. 

2322 

2372 

2423 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

7 

2981 

3031 

3082 

31:33 

3183 

3234 

3285 

3335 

3386 

3437 

8 

3487 

3538 

3589 

3639 

3690 

!  3740 

3791 

3841 

3892 

3943 

9 

3993 

4044 

4094 

4145 

4195 

|  4246 

4296 

4347 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5800 

5'JIO 

5960 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6863 

6413 

6463 

4 

6514 

6564 

6614 

6665 

6715 

67'65 

6815 

6865 

6910 

6966 

5 

7016 

7066 

7116 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819  7869 

7919 

7969 

en 

8019 

8069 

8119 

8169 

8219 

8269 

8320  !  8370 

8420 

8470 

DU 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820  8870 

8920 

8970 

9 

9020 

9070 

9120 

9170 

92fO 

9270 

9320  9369 

9419 

9469 

87'0 

9519 

9569 

9^1S 

9669 

9719 

9769 

9819  9869 

9918 

9968 

1 

940018 

0068 

0118 

0168 

0218 

0267 

0317  0367 

0417 

0467 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

6 

2504 

2t>54 

2603 

2653 

2702 

2752 

2301 

2851 

2901 

2950 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

asge 

3445 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

9 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

2 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

5 

6943 

6992 

7041 

7090 

7139 

7189 

7238 

7287 

7336 

7385 

40 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

777*7 

7826 

7875 

t±u 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

8 

8413 

8462 

8511 

8560 

8608 

8657 

8706 

8755 

8804 

8853 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

890 

9390 
9878 

9439 
9926 

9488 
9975 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

0024 

0073 

0121 

0170 

0219 

0267 

0316 

2 

950365 

0414 

0462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

3 

0&51 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

21  63 

2211 

2260 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

PROPORTIONAL  PARTS. 

Diff 

1 

2 

3      4 

5 

678 

9 

51 

5.1 

10.2 

15.3    20.4 

25.5 

30.6    35.7    40.8 

45.9 

50 

5.0 

10.0 

15.0    20.0 

25.0 

30.0    a5.0    40.0 

45.0 

49 

4.9 

9.8 

14.7    19.6 

24.5 

29.4    34.3    39.2 

44.1 

48 

4.8 

9.6 

14.4    19.2 

24.0 

28.8    33.6    38.4 

43.2 

172 


TABLU   XI. — LOGARITHMS   OF   XUMBERS. 


No  900  L.  954.1 

[No.  944  L.  975. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532  4580  4628 

4677 

1 

4725 

4773 

4821 

4869 

4918 

1866 

5014  5062  5110  5158 

2 

5207 

5255 

5.' 

503 

5351 

5399 

5447 

5495 

13  5592 

5640 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024  6072 

6120 

4 

6168 

6216 

65 

>65 

631J 

5 

6361 

6409 

6457 

65( 

>:>  r,.v,:{ 

6601 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

69 

34  7038 

7080 

48 

6 

7128 

7176 

7; 

224 

7275 

> 

7320 

7368 

7416 

74 

34 

7512 

7559 

7 

7607 

7655 

ro3 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

8 

8086 

8134 

8 

181 

822 

1 

8277 

8325 

8373 

84 

n 

84(58 

8516 

9 

8564 

8G12 

8659 

8707 

8755 

8803 

8850 

88 

e 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

1 
2 

9518 
9995 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

0042 

0090 

0138 

0185 

0233 

0280 

0328 

0376 

0423 

3 

960471 

0518 

0566 

0613 

0(561 

0709 

0756 

0804 

0851 

0899 

4 

0946 

0994 

1 

041 

108 

) 

1136 

1184 

1231 

12 

71) 

1326 

1374 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

6 

1895 

1943 

1 

990 

203 

3 

2085 

2132 

2180 

22 

'27 

2275 

2322 

7 

2369 

2417 

2 

464 

251 

1 

2559 

2606 

2653 

27 

01 

2748 

2795 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

2 

4731 

4778 

4 

K-J5 

487 

2 

4919 

4966 

5013 

5C 

161   5108 

5155 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

4 

5672 

5719 

5 

766 

581 

i 

5860 

5907 

5954 

6C 

01 

6048 

6095 

4T 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

6 

6611 

6658 

6 

705 

675 

2 

6799 

6845 

6892 

6£ 

BO 

6986 

7033 

7 

7080 

7127 

7 

173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

8 

7548 

7595 

i~ 

642 

768 

8 

7735 

7782 

7829 

7? 

75 

7922 

7969 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 
1 

8483 
8950 

8530 
8996 

8576 
9043 

8623 
9090 

8670 
9136 

8716 
9183 

8763 
9229 

8810 
9276 

8856 
9323 

8903 
9369 

2 

9416 

94(53 

9509 

9556 

9602 

9649 

9695 

B 

•42 

9789  9835 

3 

9882 

9928 

9975 

0021 

0068 

0114 

0161 

0207 

0234 

0300 

4 
5 

6 
7 
8 
9 

970347 
0812 
1276 
1740 
2203 
2666 

0393 
0858 
1322 
1786 
2249 
2712 

0440 
0904 
1369 
1832 
2295 
2758 

0486 
0951 
1415 
1879 
2342 
2804 

0533 
0997 
1461 
1925 

2388 
2851 

0579 
1044 
1508 
1971 
2434 
2897 

0626 
1090 
1554 
2018 
2481 
2943 

0672 
1137 
1601 
2064 
2527 
2989 

0719 
1183 
1647 
2110 
2573 
3035 

0765 
1229 
1693 
2157 
2619 
3082 

940 
1 
2 
3 
4 

3128 
3590 
4051 
4512 
4972 

3174 
3636 
4097 
4558 
5018 

3220 

3682 
4143 
4604 
5064 

3266 
3728 
4189 
4650 
5110 

3313 
3774 
4235 
4696 
5156 

3359 
3820 
4281 
4742 
5202 

3405 
3866 
4327 
4788 
5248 

3451 
3913 
4374 

4834 
521)4 

3497 
3959 
4420 
4880 
5340 

3543 
4005 
4466 
4926 
5386 

46 

f»ROPORTIONAL  PARTS. 

Diff.   1 

2 

3 

4 

5 

6 

7 

8 

9 

47    4.7 
46    4.6 

9.4 
9.2 

14.1 
13.8 

18.8 
18.4 

23.5 
23.0 

28.2 
27.6 

32.9 
32.2 

37.6 
36.8 

42.3 
41.4 

173 

TABLE    XT. — LOGARITHMS    OF   LUMBERS. 


No.  945  L.  975.]                                 [No.  989  L.  895. 

N. 

0 

1 

i 

a 

4 

6 

6 

7 

S 

9 

Diff. 

945 

975432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

6 

5891 

£937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

5 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

6 

C458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

7 

0912 

0957  1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

8 

1366 

1411  I  1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

•  2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

5 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5i>92 

5337 

5382 

7 

5426 

5471 

5516 

5561   5606 

5651 

5696 

5741 

5786 

58:30 

8 

5875 

5920 

5965 

6010  6055 

6100 

6144 

6189 

6234 

6279 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

3 

8113 

8157 

8202 

8247 

8291 

&336 

8331 

8425 

8470 

8514 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

6 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

9895 

9939 

9983 

0028 

0072 

0117 

0161 

0206 

0250 

0294 

8 

990a39 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1835 

1979 

2023 

2067 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

4 

2995 

3039 

3083 

3127 

3173 

3216 

3260 

3304 

3348 

3392 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

7 

4317 

4361 

4405 

4449 

4403 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

55U4 

5547 

5591 

PROPORTIONAL,  FARTS. 

Diff 

1 

23      4 

5 

678 

9 

46 

4.6 

9.2    13.8    18.4 

23.0 

27.6    32.2    36.8 

41.4 

45 

4.5 

9.0    135    18.0 

22.5 

27.0    31.5    36.0 

40.5 

44 

4.4 

8.8    13.2    17.6 

22.0 

26.4    30.8    35.2 

39.6 

43 

4.3 

8.6    12.9    17.2 

21.5 

25.8    30.1    34.4 

38.7 

174 


TABLE   XI. — LOGARITHMS   OF   NUMBERS. 


No.  990  L.  995.] 

[No.  999  L.  999. 

N. 

0 

1 

2 

3 

4         5 

6 

7 

8 

9 

Diflf. 

990 

995635 

5679 

5723 

5767 

5811      5854     5 

898 

5942 

5986 

6030 

1 

6074 

6117 

6161 

6205 

6249     6 

293     6 

337 

0380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6643 

6687     6 

"31      C 

0818 

6862 

6906 

3 

6949 

6993 

7037 

7080 

7124     7 

168     7 

212 

7255 

7299 

7343 

4 

7386 

7430 

7474 

7517 

7561      7 

305     7 

648 

7692 

7736 

7779 

5 

7823 

7867 

7910 

7954 

7998     8041     £ 

085 

8129 

8172 

8216 

6 

8259 

8303 

8347 

8390 

8434     a 

177     e 

1521 

8564 

8608 

8652 

7 

8695 

8739 

8782 

8826 

8869     8913     £ 

956 

9000 

9043 

9087 

8 

9131 

9174 

9218 

9261 

9305     9 

348     £ 

392 

9435 

9479 

9522 

9 

9565 

9609 

9652 

9696 

9739     9783     { 

1826 

9870 

9913 

9957 

43 

LOGARITHMS  OP  NUMBERS  FROM  1  TO  100. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0.000000 

21 

1.322219 

41 

1.612784 

61 

1.785330 

81 

i.noaisr. 

2 

0.301030 

22 

1.342423 

42 

1.68 

>3249 

62 

1.7£ 

12392 

82 

1.913814 

3 

0.477121 

23 

1.361728 

43 

1.633468 

63 

1.799341 

as 

.919078 

4 

0.602060 

24 

1.380211 

44 

1.6- 

13453 

64 

1.8 

)6180 

84 

.924279 

5 

0.698970 

25 

1.397940 

45 

1.653213 

65 

1.812913 

85 

.029419 

6 

0.778151 

26 

1.414973 

46 

1.662758 

66 

.819544 

86 

.934498 

7 

0.845098 

27 

1.431364 

47 

1.672098 

67 

.826075 

87 

.939519 

8 

0.903090 

28 

1.447158 

48 

1.6 

31241 

68 

.Si 

12509 

88 

.!)  His:} 

9 

0.954243 

29 

1.462398 

49 

1.690196 

69 

.a38849 

89 

.949390 

10 

1.000000 

30 

1.477121 

50 

1.698970 

70 

.845098 

90 

.954243 

11 

1.041393 

31 

1.491362 

51 

1.707570 

71 

.851258 

91 

1.959041 

12 

1.079181 

32 

1.505150 

52 

1.716003 

72 

.857332 

92 

1.1W37S8 

13 

1.113943 

33 

1.518514 

53 

1.7 

24276 

73 

.8( 

33323 

93 

1.%84S3 

14 

1.146128 

34 

1.531479 

54 

1.7 

32394 

74 

.8( 

59232 

94 

1.973128 

15 

1  176091 

35 

1.544068 

55 

1.740363 

75 

.875061 

95 

1.977724 

16 

1.204120 

36 

1.556303 

56 

1.748188 

76 

.880814 

96 

1.982271 

17 

1.230449 

37 

1.568202 

57 

1.755875 

77 

.886491 

97 

1.9K6772 

18 

1.255273 

38 

1.579784  ! 

58 

1.7 

53428 

78 

.8< 

8095 

98 

1.991226 

19 

1.278754 

39 

1.591065  ! 

59 

1.770852 

79 

.897027 

!)!> 

1  .  995635 

20 

1.301030 

40 

1.602060  ! 

60 

1.778151 

80 

.903090 

100 

2.000000 

Value   . 

at  0°.    ; 

Sign 
n  1st 
3uad. 

Vain 
at  90° 

,    .Sign 
9    in2d 
•    Quad. 

Valu 
at 

180°. 

3    Sign 
in  3d 
Quad. 

Value 
at 
270° 

Sign 
in  4th 
Quad. 

Value 
at 
360°. 

Sin  

0 

R 

4- 

O 

R 

__ 

O 

Tan  

o 

00 

o 

4- 

00. 

_ 

O 

Sec 

R 

oo 



R 

00 

4 

R 

Versin.... 

0 

R 

4- 

2R 

4. 

R 

4 

0 

Cos 

R 



o 

R 

__ 

O 

4 

R 

Cot  

00 

__ 

o 



00 

-f 

O 

_ 

00 

Cosec  

00 

-- 

R 

4- 

00 

R 

~~ 

00 

R  signifies  equal  to  rad;  oo  signifies  infinite  ;  O  signifies  evanescent. 

175 


TABLE   XII. — LOGARITHMIC   SINES, 


179° 


" 

' 

Sine. 

q-l 

Tang. 

Cotang. 

*+i 

Dl" 

Cosine. 

/ 

4.685 

15.314 

0 

0 

Inf.  neg. 

575 

575 

Inf.  neg. 

Inf.  pos. 

425 

ten 

60 

60 

1 

6.463726 

575 

575 

6.463726 

13.536274 

425 

ten 

59 

120 

2 

.764756 

575 

575 

.764756 

.235244 

425 

ten 

58 

180 

3 

6.940847 

575 

575 

6.940847 

13.059153 

425 

ten 

57 

240 

4 

7.065786 

575 

575 

7.065786 

12.934214 

425 

ten 

56 

300 

5 

.162696 

575 

575 

.162696 

.837304 

425 

ten 

55 

360 

6 

.241877 

575 

!575 

.241878 

.758122 

425 

.02 

9.999999 

54 

420 

7 

.308824 

575 

1575 

.308825 

.691175 

425 

.00 

.999999 

53 

480 

8 

.366816 

574 

1576 

.366817 

.633183 

424 

.00 

.999999 

52 

540 

g 

.417968 

574 

576 

.417970 

.582030 

424 

.00 

.999999 

51 

600 

10 

.463726 

574 

|576 

.463727 

.536273 

424 

.02 

.999998 

50 

660 

11 

7.505118 

574 

1576 

7.505120 

12.494880 

424 

.00 

9.999998 

49 

720 

12 

.542906 

574 

!  577 

.542909 

.457091 

423 

.02 

.999997 

48 

780 

13 

.577668 

574 

577 

.577672 

.422328 

423 

.00 

.999997 

47 

840 

14 

.609853 

574 

i  577 

.609857 

.390143 

423 

.02 

.999996 

46 

900 

15 

.639816 

573 

578 

.639820 

.360180 

422 

.00 

.999996 

45 

960 

16 

.667845 

573 

J578 

.667849 

.332151 

422 

.02 

.999995 

44 

1020 

17 

.694173 

573 

!578 

.694179 

.305821 

422 

.00 

.999995 

43 

1080 

18 

.718997 

573  |  !  579 

.719003 

.280997 

421 

.02 

.999994 

42 

1140 

19 

.742478 

573 

i  579 

.742484 

.257516 

421 

.02 

.999993 

41 

1200 

20 

.764754 

572 

|580 

.764761 

.235239 

420 

.00 

.999993 

40 

1260 

21 

7.785943 

572 

580 

7.785951 

12.214049 

420 

.C2 

9.999992 

39 

1320 

22 

.806146 

572 

581 

.806155 

.  193845 

419 

.02 

.999991 

38 

1380 

23 

.825451 

572 

581 

.825460 

.174540 

419 

.02 

.999990 

37 

1440 

24 

.843934 

571 

582 

.843944 

.156056 

418 

.02 

.999989 

36 

1500 

25 

.861662 

571 

583 

.861674 

.138326 

417 

.00 

.999989 

35 

1560 

26 

.878695 

571 

583 

.878708 

.121292 

417 

.02 

.999988 

34 

1620 

27 

.895085 

570  584 

.895099 

.104901 

416 

.02 

.999987 

33 

1680 

28 

.910879 

570  1  i  584 

.910894 

.089106 

416 

.02 

.999986 

32 

1740 

29 

.926119 

570 

585 

.926134 

.073866 

415 

.02 

.999985  31 

1800 

30 

.940842 

569 

586 

.940858 

.059142 

414 

.03 

.999983 

30 

1860 

31 

7.955082 

569 

!587 

7.955100 

12.044900 

413 

.02 

9.999982 

29 

1920 

32 

.968870 

569  587 

.968889 

.031111 

413 

.02 

.999981 

28 

1980 

33 

.982233 

568  588 

.982253 

.017747 

412 

.02 

.999980 

27 

2040 

34 

7.995198 

568  I  589 

7.995219 

12.004781 

411 

.02 

.999979  1  26 

2100 

35 

8.007787 

567  j  590 

8.007'809 

11.992191 

410 

.03 

.9i.;9977 

25 

2160 

36 

.020021 

567  I  591 

.020044 

.979956 

409 

.02 

.999976 

24 

2220 

37 

.031919 

566 

592 

.031945 

.968055 

408 

.02 

.999975 

23 

2280 

38 

.043501 

566 

593 

.043527 

.956473 

407 

.03 

.999973 

22 

2340 

39 

.054781 

566 

593 

.054809 

.945191 

407 

.02 

.999972 

21 

2400 

40 

.065776 

565 

:594 

.065806 

.934194 

406 

.02 

.999971 

20 

2460 

41 

8.076500 

565 

595 

8.076531 

11.923469 

405 

.03 

9.999969 

19 

2520 

42 

.086965 

564 

i596 

.086997 

.913003 

404 

.02 

.999968 

18 

2580 

43 

.097183 

564 

J598 

.097217 

.902783 

402 

.03 

.999966 

17 

2640 

44 

.107167 

563 

1599 

.107203 

.892797 

401 

.03 

.999964 

16 

2700 

45 

.116926 

562 

1600 

.116963 

.883037 

400 

.02 

.999963 

15 

2760 

46 

.126471 

562 

|601 

.126510 

.873490 

399 

,03 

.999961 

14 

2820 

47 

.135810 

561 

602 

.135851 

.864149 

398 

.03 

.999959 

13 

2880 

48 

.144953 

561 

J603 

.144996 

.855004 

397 

.02 

.999958 

12 

2940 

49 

.153907 

560 

604 

.153952 

.846048 

396  |  «5g 

.999956 

11 

3000 

5C 

.162681 

560 

1605 

.162727 

.837273 

395  1  -03 

.999954 

10 

3060 

51 

8.171280 

559 

J607 

8.171328 

11.828672 

393 

•  .03 

9.999952 

9 

3120 

52 

.179713 

558  608 

.179763 

.820237 

392 

1  .03 

.999950 

8 

3180 

53 

.187985 

558  |  i  609 

.188036 

.811964 

391 

.03 

.999948 

7 

3240 

54 

.196102 

557 

611 

.196156 

.803844 

389 

.('3 

.999946 

6 

3300 
3360 

55 
56 

.204070 
.211895 

556 
556 

612 
1613 

.204126 
.211953 

.795874 

.788047 

388 
387 

.03 
.03 

.899944 
.999942 

5 
4 

8120 

57 

.219581 

555  i  615 

.219641 

.780359 

385 

.03 

.999940 

3 

3480 

58 

.227134 

554  616 

.227195 

.772805 

384 

.03 

.999938 

2 

3540 

59 

.234557 

554 

618 

.234621 

.765379 

382 

.03 

.999936 

1 

3600 

60 

8.241855 

553 

619 

8.241921 

11.758079 

381 

.03 

9.999934 

0 

4.685 

15.314 

" 

t 

Cosine. 

q-l 

Cotang. 

Tang. 

ff-M 

Dl" 

Sine. 

' 

90° 


176 


89- 


OSINES,   TANGENTS,   AND    COTANGENTS. 


178° 


" 

' 

Sine. 

i-i 

Tang. 

Cotang. 

q  +  l 

Dr 

Cosine. 

, 

I  4.685 

15.314 

1 

3600 
3660 
3720 
3780 
3840 
3900 
3960 
4020 
4080 
4140 

1 

2 
3 
4 
5 

6 

7 
8 
9 

8.241855 
.249033 
.  256094 
.263042 

!  .269881 
.276614 
.283243 
i  .289773 
.296207 
.  302546 

553 
552 
551 
551 
550 
549 
548 
547 
546 
546 

019 
020 
022 
623 
625 
627 
628 
630 
032 
033 

8.241921 
.24910-2 
.250105 
.203115 
.269956 
.276691 
.283323 
.289856 
.296292 
.302634 

11.  7-58070 
.750898 

.786885 

.730044 
.723309 
.710077 
.710144 
.7037'08 
.697866 

381 
380 
378 
377 
375 

372 
370 
368 
367 

.oa 

.06 
.08 
.08 
.06 

.03 

.05 

.03 

9.999931 
.999988 

.99998! 

.999985 

.999915 
999918 

00 
59 
58 
57 
50 
55 
54 
58 
52 
51 

4200   10   .308794  545 

635 

,308884 

.691116 

365 

.05 

!  999910 

5C 

4260 
4320 
4380 
4440 
4500 
4560 
4620 
4680 

11 
12 
13 
14 
15 
16 
17 
18 

8.314954 
.321027 
.32701(5 
.332924 
.338753 
.344504 
.350181 
.355783 

544 
543 

542 
541 
540 

539 
539 
538 

637 
638 
640 
642 
644 
646 
648 
649 

8.315046 
.321122 
.327114 
.333025 
.338856 
.344610 
.350289 
.355895 

11.684954 
.67'8878 
.672886 
.666975 
.661144 
.655390 
.649711 
.644105 

363 
302 
360 
358 
356 
354 
352 
351 

.05 
.03 
.05 
.05 
.03 
.05 
.05 
.05 

9.999907 
.999906 
.999908 
.999899 

48 
47 
46 

43 
42 

4740 

19 

.361315 

537 

651 

.361430 

.638570 

349   •"•? 

QQQRBK 

41 

4800 

20 

.366777 

536 

653 

.366895 

.633105 

347   'Uo   .9WW 

40 

4860 
4920 
4980 

21 
22 
23 

8  372171 
.377499 

.382762 

535 
534 
533 

655 
657 
659 

8.372292 
.377622 
.382889 

11.627708 
.622378 
.617111 

345  a! 

.'543  •;;? 

341   'SS 

9.999873 
.999876 

.999873 

38 

5040 

24 

.387962 

532 

661 

.388092 

.611908 

339   •<*? 

30 

5100 

25 

.393101 

531 

663 

.393234 

.606766 

337  ;  'SB 

.999^07 

35 

5160 

26 

.398179 

530 

666 

.398315 

.601685 

334 

.uo 

.999664 

34 

5220 

27 

.403199 

529 

668 

.403338 

.596662 

332 

.05 

.999861 

33 

5280 

28 

.408161 

527 

670 

.408304 

.591696 

330 

t  .05 

.998858 

32 

5340 

29 

.413068 

526 

672 

.413213 

.586787 

328 

.07 

.998854 

31 

5400 

30 

.417919 

525 

674 

.418068 

.581932 

326 

.05 

.999851 

30 

5460 

31 

8.422717 

524 

676 

8.422869 

11.577131 

324 

.05 

9.90HK 

5520 

32 

.427462 

523 

679 

.427618 

.572382 

321 

.07 

5580 

33 

.432156 

522 

681 

.432315 

.567685 

319 

.05 

5640 

34 

.436800 

521 

683 

.436962 

.563038 

317 

.05 

18  !j> 

5700 

35 

.441394 

520 

685 

.441560 

.558440 

315 

.07 

.99989 

5760 

36 

.445941 

518 

688 

.446110 

.553890 

312 

.05 

.999881  24 

5820 

37 

.450440 

517 

690 

.450613 

.549387 

310 

.07 

.998W7  23 

5880 

38 

.454893 

516 

693 

.455070 

.544930 

307 

.05 

5940 

39 

.459301 

515 

695 

.459481 

.540519 

305 

.07 

.99982d  21 

6000 

40 

.463665 

514 

697 

.463849 

.536151 

303 

.07 

.999810  20 

6060 

41 

8.467985 

512 

700 

8.468172 

11.531828 

300 

.05 

9.998813  11) 

6120 

42 

.472263 

511 

702 

.472454 

.527546 

298 

. 

6180 

43 

.476498 

510 

705 

.476693 

.523307 

295 

.07 

.998N 

6240 

44 

.480693 

509 

707 

.480892 

.519108 

293 

.07 

91  i  10 

6300 

45 

.484848 

50v' 

710 

.485050 

.514950 

290 

.07 

.tt9?tf 

6360 

46 

.488963 

506 

713 

.489170 

.510830 

287 

.05 

/\r- 

.999794   14 

6420 

47 

.493040 

505 

715 

.493250 

.506750 

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•JE   .99MJ.O   13 

6480 

48 

.497078 

503 

718 

.497293 

.502707 

888 

•}{i   .9897HJ   12 

6540 

49 

.501080 

502 

720 

.501298 

.498702 

280 

.UY 

-2   11 

6600 

50 

.505045 

501 

723 

.505267 

.494733 

277 

•  ' 

.999778  10 

6660 

51 

8.508974 

499  ' 

726 

8.509200 

11  .490800 

274 

.07  8.c,n<)774   '.» 
.08 

6720 

52 

.512867 

498 

729 

.513098 

.486902 

271 

.99974 

6780 

53 

.516726 

497 

731 

.510901 

.483039 

269 

•  ' 

6840 

54 

.520551 

495 

734 

.520790 

.479210 

266 

*07 

.999761 

6 

6000 

55 

.524343 

494 

737 

524586 

.475414 

263 

.999767 

5 

6960 

56 

.528102 

492 

740 

.528349 

.471651 

260 

*08 

.999768 

4 

7020 

57 

.531828 

491 

743 

.532080 

.467920 

257 

*07 

.999748 

3 

7080 

58 

.535523 

490 

745 

.5:35779 

.464221 

255 

*07 

.999744 

2 

7140 

59 

.539186 

488 

748 

.539447 

.460553 

252 

•08 

.999740 

1 

7200 

60 

8.542819 

487 

751 

8.543084  1 

11.456916 

249 

9.990786 

0 

4.685 

15.314 

~"~~ 

~ 

Cosine. 

q-l 

Cotang. 

Tang. 

q  +  l 

Dr 

Sine. 

'j 

91° 


177 


88 


TABLE   XII. — LOGARITHMIC   SINES 


177° 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 

8  542319 
.546422 
.549995 
.553539 
.557054 
.560540 
.563999 
.567431 

60.05 
59.55 
59.07 
58.58 
58.10 
57.65 
57.20 

KR  i"K 

9.999735 
.999731 
.999726 
.999722 
.999717 
.999713 
.999708 
.999704 

.07 
.08 
.07 
.08 
.07 
.08 
.07 

AQ 

8.5430S4 
.546691 
.550268 
.553817 
.557336 
.560828 
.564291 
.567727 

60.12 
59.62 
59.15 
58.65 
58.20 
57.72 
57.27 

t£»  OQ 

11.456916 
.453309 
.449732 
.446183 
.442664 
.439172 
.435709 
.432273 

60 
59 
58 
57 
56 
55 
54 
53 

8 
9 

.570836 
.574214 

OO.  <O 

56.30 

.999699 
.999694 

.Uo 

.08 

.571137 
.574520 

OO.oO 

56.38 

.428863 
.425480 

52 
51 

10 

.577566 

55.87 
55.43 

.999689 

.08 
.07 

.577877 

55  .  95 
55.52 

.422123 

50 

11 
12 
13 
14 
15 
16 

8.580892 
.584193 
.587469 
.590721 
.593948 
.597152 

55.02 
54.60 
54.20 
53.78 
53.40 

tO  AT) 

9.999685 
.999680 
.999675 
.999670 
.999665 
.999660 

.08 
.08 
.08 
.08 
.08 

AQ 

8.581208 
.584514 
.587795 
.591051 
.594283 
,597492 

55.10 
54.68 
54.27 
53.87 
53.48 

tO  AQ 

11.418792 
.415486 
.412205 
.408949 
.405717 
.402508 

49 

48 
47 
46 
45 
44 

17 
18 
19 

.600332 
.603489 
.606623 

Do.UU 

52.62 
52.23 

.999655 
.999650 
.999645 

.Uo 

.08 
.08 

.600677 
.603839 
.606978 

oo.Uo 
52.70 
52.32 

.399323 
.396161 
.393022 

43 
42 
41 

20 

.609734 

51.85 
51.48 

.999640 

.08 
.08 

.610094 

51  .93 
51.58 

.389906 

40 

21 
22 
23 
24 

8.612823 
.615891 
.618937 
.621962 

51.13 
50.77 
50.42 

9.999635 
.999629 
.999624 
.999619 

.10 
.08 
.08 

8.613189 
.616262 
.619313 
.622343 

51.22 
50.85 
50.50 

11.386811 
.383738 
.380687 
.377657 

39 

38 
37 
36 

25 

26 
27 
28 
29 
30 

.624965 
.627948 
.630911 
.633854 
.636776 
.639680 

50.05 
49.72 
49.38 
49.05 

48.70 
48.40 
48.05 

.999614 
.999608 
.999603 
.999597 
.999592 
.999586 

.08 
.10 
.08 
.10 
.08 
.10 
.08 

.625352 
.628340 
.631308 
.634256 
.637184 
.640093 

50.15 
49.80 
49.47 
49.13 
48.80 
48.48 
48.15 

'  .374648 
.371660 
.368692 
.365744 
.362816 
.359907 

35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.642563 
.645428 
.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 
.667689 

47.75 
47.43 
47.13 
46.82 
46.52 
46.22 
45.92 
45.63 
45.35 
45.07 

9.999581 
.999575 
.999570 
.999564 
.999558 
.999553 
.999547 
.999541 
.999535 
.999529 

.10 
.08 
.10 
.10 
.08 
.10 
.10 
.10 
.10 
.08 

8.642982 
.645853 
.648704 
.651537 
.654352 
.657149 
.659928 
.662689 
.6654.33 
.668160 

47.85 
47.52 
47.22 
46.92 
46.62 
46.32 
46.02 
45.73 
45.45 
45.17 

11.357018 
.354147 
.351296 
.348463 
.345648 
.342851 
.340072 
.337311 
.334567 
.331840 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 

8.670393 
.673080 
.675?'51 

.678405 
.681043 
.6a3665 
.686272 
.688863 
.6914*^8 

44.78 
44.52 
44.23 
43.97 
43.70 
43.45 
43.18 
42.92 

9.999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 
.999481 
.999475 

.10 
.10 
.10 
.10 
.12 
.10 
.10 
.10 

1ft 

8.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 
.691963 

44.88 
44.60 
44.35 
44.07 
43.80 
-43.53 
43.28 
43.03 

11.329130 
.326437 
.323761 
.321100 
.318456 
.315828 
.313216 
.310619 
.308037 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.693998 

42.67 

42.42 

.999469 

.  1U 

.10 

.694529 

42.77 
42.53 

.305471 

10 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

8.696543 
.699073 
.701589 
.704090 
.706577 
.709049 
.711507 
.713952 
.716383 
8.718800 

42.17 
41.93 
41.68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.28 

9.999463 
.999456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999418 
.999411 
9.999404 

.12 
.10 
.12 
.10 
.10 
.12 
.10 
.12 
.12 

8.697081 
.699617 
.702139 
.704646 
.707140 
.709618 
.712083 
.714534 
.716972 
8.719396 

42.27 
42.03 
41.78 
41.57 
41.30 
41.08 
40.85 
40.63 
40.40 

11.302919 

.300383 
.297861 
.295354 
.292860 
.290382 
.287917 
.285466 
.283028 
11.280604 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

r 

Ooe^e. 

D.  r.  i 

Sine.  !  D.  1".  1 

Cotang. 

D.  1'. 

Tang. 

r 

92' 


178 


COSINES,    TANGENTS,    AND    COTANGENTS. 


176° 


' 

Sine. 

D.I". 

Cosine. 

D.  r. 

Tang. 

D.  I'. 

Cotang. 

- 

0 

1 

2 
3 
4 
5 

6 

8 
9 

8.718800 
.721204 
.723595 
.725972 
.728337 
.730688 
.733027 
.7&53S4 
.737667 
.739969 

40.07 
39.85 
39.62 
39.42 
39.18 
38.98 
38.78 
38.55 
38.37 

9.999404 
.999398 
.999391 
.91)9384 
.999378 
.999371 
.999364 
.999357 
.999350 
.999343 

.10 
.12 
.12 
.10 
.12 
.12    i 
.12    i 
.12 
.12 

8.719306 
.721806 
.724204 
.7205S8 
.728959 
.731317 
.733663 
.735996 
.738317 
.740626 

40.17 
89.97 
89.78 
89.52 
89.80 
39.10 
38.88 
38.68 
88.48 

11.280604 
,278194 
.275796 
.278412 
.271041 

.266887 
364004 
.261668 
.259874 

60 

59 
58 

r.7 

56 
59 
54 

53 

u 

51 

10 

.742259 

37.95 

.999336 

!l2    i      -7429;^ 

38.08 

.257078 

50 

11 
12 

8.744536 
.746802 

37.77 

9.999329 

.999322 

.12 

8.745207 
.747479 

37.87 

!  252521 

49 

48 

13 
14 
15 
16 
17 
18 
19 
20 

.749055 
.751297 
.753528 
.755747 
.757955 
.760151 
.7623137 
.764511 

37.37 
37.18 
36.98 
36.80 
36.60 
36.43 
36.23 
36.07 

.999315 
.999308 
.999301 
999294 
.999287 
.999279 
.999272 
.999265 

.12 
.12 
.12 
.12 
.13 
.12 
.12 
.13 

.749740 
.751989 
.754227 
.756453 
.758668 
.760872 
.763065 
.765246 

37.48 
37.30 
37.10 
36.92 
36.73 
36.55 
86.85 
36.18 

.250260 
.248011 
.245773 
.248547 

!2391:iS 
.286985 
.284754 

47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 

8.766675 
.768828 
.770970 
.773101 
.775223 

35.88 
35.70 
35.52 
35.37 

9.999257 
.999250 
.999242 
.999235 
.999227 

.12 
.13 
.12 
.13 

8.767417 
.769578 
.771727 
.773866 
.775995 

86  JOB 

as.  82 

a5.65 
85.48 

oe   oo 

11.232583 
.230422 
.228273 
.226134 
.224005 

39 
38 
371 
36 
35 

26 
27 
28 
29 
30 

.777333 
.779434 
.781524 
.783605 
.785675 

35.17 
35.02 
34.83 
34.68 
34.50 
34  35 

.999220 
.999212 
.999205 
.999197 
.999189 

.12 
.13 
.12 
.13 
.13 
.13 

.778114 
.780222 
i     .782320 
.784408 
.786486 

35.13 
34.97 
34.80 
34.63 
34.47 

.221886 
.219778 
.217680 
.215592 
.213514 

34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.787736 
.789787 
.791828 
.793859 
.795881 
.797894 
.799897 
.801892 
.803876 
.805852 

34.18 
34.02 
33.85 
33.70 
33.55 
33.38 
33.25 
33.07 
32.93 
3°  78 

9.999181 
.999174 
.999166 
.999158 
.999150 
.999142 
.999134 
.999126 
.999118 
.999110 

.12 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

8.788554 
.790613 
.792662 
.794701 
.796731 
.796752 
.800763 
.802765 
.804758 
.8067'42 

34.32 
34.15 
33.98 

as.  as 
as.  68 

33.52 

as.  37 

33.22 

as.  07 

32.92 

11.211446 
.209387 
.207338 
.205299 
.203269 
.201248 
.199237 
.197235 
.195242 
.193258 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

8.807819 
.809777 
.811726 
.813667 
.815599 
.817522 
.819436 
.821343 
.823240 
.825130 

32.63 
32.48 
32.  a5 
32.20 
32.05 
31.90 
31.78 
31.62 
31.50 
31  35 

9.999102 
.999094 
.999086 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 
.999027 

.13 
.13 
.15 
.13 
.13 
.13 
.15 
.13 
.15 
13 

8.808717 
.810683 
.812641 
.814589 
.816529 
.818461 
.820384 
.822298 
.824205 
.826103 

88.W 

32.63 
32.47 
32.33 
32.20 
32.05 
31.90 
31.78 
31.63 
31.48 

11.191283 
.189317 
.187359 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 
.173897 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.827011 
.828884 
.a30749 
.832607 
.834456 
.836297 
.838130 
.839956 
.841774 
8.843585 

31.22 
31.08 
30.97 
30.82 
30.68 
30.55 
30.43 
30.30 
30.18 

9.999019 
.999010 
.999002 
.998993 
.998984 
.998976 
.998967 
.998958 
.998950 
9.998941 

.15 
.13 
.15 
.15 
.13 
.15 
.15 
.13 
.15 

8.827992 
.829874 
.831748 
.833613 
.835471 
.837321 
.839163 
.840998 
.842825 
8.844644 

31.37 
31.23 
31.08 
30.97 
30.83 
30.70 
30.58 
30.45 
30.32 

11.172008 
.170126 
.166852 
.166867 

.164529 
.162679 
.160687 
.159002 
.157175 
11.155356 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

/ 

Cosine. 

D  1*. 

Sine. 

D.  I'. 

Cotang. 

D.  r.        Tang. 

L_  86- 

179 


TABLE   XII. — LOGARITHMIC    SINES, 


174° 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1\ 

Tang. 

D.  1". 

Cotang. 

t 

0 

1 

2 
3 
4 
5 
6 

8 
9 
10 

8.8435a5 
.845387 
.847183 
.848971 
.850751 
.852525 
.854291 
.856049 
.857801 
.859546 
.861283 

30.03 
29.93 

29.80 
29.67 
29.57 
29.43 
29.30 
29.20 
29.08 
28.95 
28.85 

9.998941 
.998932 
.998923 
.998914 
.998905 
.998896 
.998887 
.998878 
.998869 
.998860 
.998851 

.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.17 

8.844644 
.846455 
.848260 
.850057 
.851846 
.853628 
.855403 
.857171 
.858932 
.860686 
.862433 

30.18 
30.08 
29.95 
29.82 
29.70 
29.58 
29.47 
29.35 
29.23 
29.12 
29.00 

'.  153545 
.151740 
.149943 
.148154 
.146372 
.144597 
.142829 
.141068 
.139314 
.137567 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 

8.863014 
.864738 
.866455 
.868165 
.869868 
.871565 

28.73 
28.62 
28.50 
28.38 

28.28 

90  -\rt 

9.998841 
.998832 
.998823 
.998813 
.998804 
.998795 

.15 
.15 
.17 
.15 
.15 

8.864173 
.865906 
.867632 
.869351 
.871064 
.872770 

28.88 
28.77 
28.65 
28.55 
28.43 

no  oo 

11.135827 
.134094 
.  132368 
.130649 
.128936 
.127230 

49 
48 
47 
46 
45 
44 

17 

18 
19 
20 

.873255 
.874938 
.876615 

.878285 

tCO.  i  i 

28.05 
27.95 
27,83 
27.73 

.998785 
.998776 
.998766 
.998757 

.  5 

*  k 

.874469 
.876162 
.877849 
.879529 

/CO  .  Ofi 

28.22 
28.12 
28.00 
27.88 

.  125531 
.123838 
.122151 
.120471 

43 
42 
41 
40 

21 
22 

8.879949 
.881607 

27.63 

9.998747 
.998738 

.  5 

•  n 

8.881202 
.882869 

27.78 

11.118798 
.117131 

39 

38 

23 
24 
25 

.883258 
.884903 

.886542 

27.52 
27.42 
27.32 

.998728 
.998718 
.998708 

K 

.884530 

.886185 
.887833 

27!  58 
27.47 
27  38 

.115470 
.113815 
.112167 

37 
36 
35 

26 

.888174 

/«<  •  "" 

.998699 

.  •) 

.889476 

.110524 

34 

27 

.889801 

27.12 

.998689 

•  ^ 

.891112 

27.27 

.108888 

33 

28 

.891421 

27.00 

.998679 

'  i 

.892742 

27.17 

.107258 

32 

29 

.893035 

26.90 

.998669 

-  ^ 

.894366 

27.07 

.105634 

31 

30 

.894643 

26.80 
26.72 

.998659 

;  f 

.895984 

26.97 
26.87 

.104016 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.896246 
.897842 
.899432 
.901017 
.902596 
.904169 
.905736 
.907297 
.908853 
.910404 

26.60 
26.50 
26.42 
26.32 
26.22 
26.12 
26.02 
25.93 
25.85 
25.75 

9.998649 
.998639 
.998629 
.998619 
.998609 
.998599 
.998589 
.998578 
.998568 
.998558 

.17 
.17 
.17 
.17 
.17 
.17 
.18 
.17 
.17 
.17 

8.897596 
.899203 
.900803 
.902398 
.903987 
.905570 
.907147 
.908719 
.910285 
.911846 

26.78 
26.67 
26.58 
26.48 
26.38 
26.28 
26.20 
26.10 
26.02 
25.92 

11.102404 
.100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091281 
.089715 
.088154 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 

8.911949 
.913488 
.915022 

25.65 
25.57 

9.998548 
.998537 
.998527 

.18 
.17 

8.913401 
.914951 
.916495 

25.83 
25.73 
25  63 

11.086599 
.085049 
.083505 

19 
18 
17 

44 
45 
46 
47 
48 
49 
50 

.916550 
.918073 
.919591 
.921103 
.922610 
.924112 
.925609 

25  .'38 
25.30 
25.20 
25.12 
25.03 
24.95 
24.85 

.998516 
.998506 
.998495 
.998485 
.998474 
.998464 
.998453 

'i? 

.18 
.17 
.18 
.17 
.18 
.18 

.918034 
.919568 
.921096 
.922619 
.924136 
.925649 
.927156 

25!  57 
25.47 
25.38 
25.28 
25.22 
25.12 
25.03 

.081966 
.080432 
.078904 
.077381 
.075864 
.074351 
.072844 

16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.927100 
.928587 
.930068 
.931544 
.933015 
.9:34481 
.935942 
.937398 
.938850 
8.940296 

24.78 
24.68 
24.60 
24.52 
24.43 
24.35 
24.27 
24.20 
24.10 

9.998442 
.998431 
.998421 
.998410 
.998399 
.998388 
.998377 
.998366 
.998355 
9.998344 

.18 
.17 
.18 
.18 
.18 
.18 
.18 
.18 
.18 

8.928658 
.930155 
.931647 
.933134 
.934616 
.936093 
.937565 
.939032 
.940494 
8.941952 

24.95 

24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.30 

.11.071342 
.069845 
.068353 
.066866 
.065384 
.063907 
.062435 
.060968 
.059506 
11.058048 

9 
8 
7 
6 
5 

3 

2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1'. 

Cotang. 

D.I". 

Tang. 

' 

94' 


180 


85* 


COSINES,    TANGENTS,    AND    COTANGENTS.  174° 


' 

Sin«. 

D.  1".  !  Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

• 

0 

1 

2 

3 
4 
5 
6 
7 
8 
9 
10 

8.940296 
.941738 
.943174 
.944606 
.946034 
.947456 
.948874 
.950287 
.951696 
.953100 
.954499 

24.03 
88.93 

23.87 
23.80 
23.70 
23.63 
23.55 
23.48 
23.40 
23.32 
23.25 

1  9.998344 

!  998322 
.998311 
.998300 
.998289 
.998277 
.998266 
.998255 
.998243 
|  .998232 

.18 
.18 
.18 

.18 
.18 
.20 
.18 
.18 
.20 
.18 
.20 

!  948404 

.944852 
.946295 
.947734 
.949168 
.950697 
.952021 
.953441 
.954856 

24.20 
84.18 
84.05 

23.98 
23.90 
23.82 
23.73 
23.67 
23.58 
23.52 
23  45 

11.058048 
.066698 

.055148 
.053705 

.060888 

.049-403 
.047979 
.046569 
.045144 
.043733 

60 
59 
68 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 

13 
14 
15 
16 
17 
18 
19 
20 

8.955894 
.957284 
.958670 
.960052 
.961429 
.962801 
.964170 
.965534 
.966893 
.968249 

23.17 
23.10 
23.03 
22.95 

22.87 
22.82 
22.73 
22.65 
22.60 
22.52 

9.998220 
.998209 
.998197 
.998186 
.998174 
.998163 
.998151 
.998139 
.998128 
.993116 

.18 
.20 
.18 
.20 
.18 
.20 
.20 
.18 
.20 
.20 

8.957674 
.959075 
.960473 
.961866 
.963255 
.964639 
.966019 
.967394 
.968766 
.970133 

23.35 
23.30 
23.22 
23.15 
23.07 
23.00 
22.92 
22.87 
22.78 
22  72 

11.042326 
.040985 
.039527 
.088184 

.036745 
.035361 
.033981 
.032606 
.081884 
.029867 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

8.969600 
.970947 
.972289 
.973628 
.974962 
.976293 
.977619 
.978941 
.980259 
.981573 

22.45 
22.37 
22.32 
22.23 
22.18 
22.10 
22.03 
21.97 
21.90 
21.83 

9.998104 
.998092 
.998080 
.998068 
.998056 
.998044 
.998032 
.998020 
.998008 
.997996 

.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 

8.971496 
.972855 
.974209 
.975560 
.976906 
.978248 
.979586 
.980921 
.982251 
.983577 

22.65 
22.57 
22.52 
22.43 
22.37 
22.30 
22.25 
22.17 
22.10 
22.03 

11.028504 
.027145 
.025791 
.024440 
.023094 
.021752 
.020414 
.019079 
.017749 
.016423 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.982883 
.984189 
.985491 
.986789 
.988083 
.989374 
.990660 
.991943 
.993222 
.994497 

21.77 
21.72 
21.63 
21.57 
21.52 
21.43 
21.38 
21.32 
21.25 
21.18 

9.997984 
.997972 
.997959 
.997947 
.997935 
.997922 
.997910 
.997897 
.997885 
.997872 

.20 
.22 
.20 
.20 
.22 
.20 
.22 
.20 
.22 
.20 

8.984899 
.986217 
.987532 
.988842 
.990149 
.991451 
.992750 
.994045 
.995337 
.996624 

21.97 
21.92 
21.83 
21.78 
21.70 
21.65 
21.58 
21.53 
21.45 
21.40 

11.015101 
.013783 
.012468 
.011158 
.009851 
.008549 
.007250 
.005955 
.004663 
.003376 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 

8.995768 

01  i  <* 

9.997860 

oo 

8.997908 

91  3H 

11.002092 

19 

42 

.997036 

.997847 

on 

8.999188 

01  oo 

11.000812 

18 

43 
44 
45 

46 
47 
48 
49 
50 

.998299 
8.999560 
9.000816 
.002069 
.003318 
.004563 
.005805 
.007044 

21.02 
20.93 
20  88 
20.82 
20.75 
20.70 
20.65 
20.57 

.997835 
.997822 
.997809 
.997797 
.997784 
.997771 
.997758 
.997745 

.22 
.22 
.20 
.22 
.22 
.22 
.22 
.22 

9.000465 
.001738 
.003007 
.004272 
.005534 
.006792 
.008047 
.009298 

21.22 
21.15 
21.08 
21.03 
20.97 
20.92 
20.85 
20.80 

10.999535 
.998262 
.996993 
.995728 
.994406 
.993208 
.991953 
.990702 

17 
16 
15 
14 
13 
12 
11 
10 

51 
52 

9.008278 
.009510 

20.53  ! 

9.997732 
.997719 

.22 

9.010546 
.011790 

20.73 

on  «a 

10.989454 
.988840 

9 

8 

53 
64 
55 
56 
57 
58 
59 
60 

.010737 
.011962 
.013182 
.014400 
.015613 
.016824 
.018031 
9.019235 

20.42 
20.33 
20.30 
20.22 
20.18 
20.12 
20.07 

.997706 
.997693 
.997680 
.997667 
.997654 
.997641 
.997628 
9.997614 

.22 
.22 
.22 
.22 
.22  i 
.22  1 
.23  i 

.013031 
.014268 
.015502 
.016732 
.017959 
.0191&3 
.020403 
9.021620 

20.62 
20  57 
20.50 
20.45 
20.40 
20.33 
20.28 

.986969 
.986788 
.984498 

.988868 
.988041 
.980617 
.979697 

10.978380 

7 
6 
5 
4 
3 
2 
1 
0 

• 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

1 

05° 


181 


81* 


TABLE   XII. — LOGARITHMIC    SINES, 


173° 


' 

Sine. 

D.  1'. 

Cosine. 

D.I-. 

Tang. 

D.  r. 

Cotang. 

• 

0 

1 

2 
3 
4 
5 
6 

8 
9 
10 

9.019235 
.020435 
.021632 
.022825 
.024016 
.025203 
.026386 
.027567 
.028744 
.029918 
.031089 

20.00 
19.95 
19.88 
19.85 
19.78 
19.72 
19.68 
19.62 
19.57 
19  52 
19.47 

9.997614 
.997601 
.997588 
.997574 
.997561 
.997547 
.997534 
.997520 
.997507 
.997493 
.997480 

.22 
.22 

.23 

!23 
.22 
.23 
.22 
.23 
22 

9.021620 
.022834 
.024044 
.025251 
.026455 
.027655 
.028852 
.030046 
.031237 
.032425 
.033609 

20.23 
20.17 
20.12 
20.07 
20.00 
19.95 
19.90 
19.85 
19.80 
19.73 
19.70 

10.978380 
.977166 
.975956 
.974749 
.973545 
.972345 
.971148 
.969954 
.968763 
.967575 
.966391 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.032257 
.033421 
.034582 
.035741 
.036896 
.038048 
.039197 
.040342 
.041485 
.042625 

19.40 
19.35 
19.32 
19.25 
19.20 
19.15 
19.08 
19.05 
19.00 
18.95 

9.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 
.997341 

.23 
.22 
.23 
.23 
.23 

!23 
.23 
.23 

.23 

9.034791 
.035969 
.037144 
.038316 
.039485 
.040651 
.041813 
.042973 
.044130 
.045284 

19.63 
19.58 
19.53 
19.48 
19.43 
19.37 
19.33 
19.28 
19.23 
19.17 

10.965209 
.964031 
.962856 
.961684 
.960515 
.959349 
.958187 
.957027 
.955870 
.954716 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 

9.043762 
.044895 
.046026 
.047154 

18.88 
18.85 
18.80 

9.997327 
.997313 
.997299 
.997285 

.23 
.23 
.23 

9.046434 
.047582 

.048727 
.049869 

19.13 
19.08 
19.03 

18  Oft 

10.953566 
.952418 
.951273 
.950131 

39 
38 
37 
36 

25 

26 
27 

28 

.048279 
.049400 
.050519 
.051635 

18.75 
18.68 
18.65 
18.60 

.997271 
.997257 
.997242 
.997228 

.23 
.23 
.25 
.23 

.051008 
.052144 
.053277 
.054407 

Jo.yo 
18.93 
18.88 
18  83 

18  8ft 

.948992 
.947856 
.946723 
.945593 

35 
34 
33 
32 

29 
30 

.052749 
.053859 

18.57 
18.50 
18.45 

.997214 
.997199 

.23 
.25 
.23 

.055535 
.056659 

Jo.oU 

18.73 
18.70 

,944465 
.943341 

31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

9.054966 
.056071 
.057172 
.058271 
.059367 
.060460 
.061551 
.062639 
.063724 
.064806 

18.42 
18.35 
18.32 
18.27 
18.22 
18.18 
18.13 
18.08 
18.03 
17.98 

9.997185 
.997170 
.997156 
.997141 
.997127 
.997112 
.997098 
.997083 
.997068 
.997053 

.25 
.23 
.25 
.23 
.25 
•  .23 
.25 
.25 
.25 
.23 

9.057781 
.058900 
.060016 
.061130 
.062240 
.063348 
.064453 
.065556 
.066655 
.067752 

18.65 
18.60 
18.57 
18.50 
18.47 
18.42 
18.38 
18.32 
18.28 
18.25 

10.942219 
.941100 
.939984 
.938870 
.937760 
.936652 
.935547 
.934444 
.933345 
.932248 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 

9.065885 
.066962 
.068036 
.069107 
.070176 
.071242 

17.95 
17.90 
17.85 
17.82 

1H7 

9.997039 
.997024 
.997009 
.996994 
.996979 
.996964 

.25 
.25 
.25 
.25 
.25 

9.068846 
.069938 
.071027 
.072113 
.073197 
.074278 

18.20 
18.15 
18.10 
18.07 
18.02 

10.931154 
.930062 
.928973 
.927887 
.926803 
.925722 

19 
18 
17 
16 
15 
14 

47 

.072306 

<•»  A7 

.996949 

.25 

|  .075356 

17.97 

.924644 

13 

48 

.073366 

17*  AQ 

.996934 

.25 

.076432 

17.93 

.923568 

12 

49 
50 

.074424 
.075480 

li  .OO 

17.60 
17.55 

.996919 
.996904 

.25 
.25 
.25 

.077505 
.078576 

17.88 
17.85 
17.80 

.922495 
.921424 

11 
10 

51 
52 
53 
54 

9.076533 
.077583 
.078631 
.079676 

17.50 

17.47 
17.42 

1r»  QQ 

9.996889 
.996874 
.996858 
.996843 

.25 
.27 
.25 

9.079644 

.080710 
.081773 

.082833 

17.77 
17.72 
17.67 

10.920356 
.919290 
.918227 
.917167 

9 

8 
7 
6 

55 
56 

.080719 
.081759 

<  .  oo 

17.33 

.996828 
.996812 

.27 
27 

.083891 
.084947 

17  '.60 

.916109 
.915053 

5 
4 

57 

58 

.082797 
.083832 

17.30 
17.25 

.996797 
.996782 

'.25 

.086000 
.087050 

17.55 
17.50 

17  /17 

.914000 
.912950 

3 
2 

59 

.084864 

17.20 

.996766 

•~* 

.OS8098 

(  .4< 

.911902 

1 

60 

9.085894 

17.17 

9.996751 

.2o 

9.089144 

17.43 

10.910856 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang.  |  D.  1".  |  Tang. 

' 

98° 


182 


83* 


COSINES,    TANGENTS,    AND    COTANGENTS.  172° 


• 

Sine. 

D.  1'. 

Cosine. 

B 

Tan,. 

D.  r. 

Cotang. 

/ 

0 

1 

2 

9  085894 
.086922 

.087947 

17.13 

17.08 

9.996751 
.  990735 
990?'20 

27 

9.089144 
.090187 

17.38 

17  :>5 

10.910856 
.909818 

60 
59 

3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

.  088970 
.089990 
.091008 
.092024 
.093037 
.094047 
.095056 
.096062 

9  097065 
.098066 
.  099065 
.  100002 
.101056 
.  102048 
.  103037 
.  104025 
.105010 
.  105992 

17.00 
10.97 
10.93 
10.88 
16.83 
16.82 
16.77 
16.72 

16.68 
16.65 
16.62 
16.57 
16.53 
16.48 
16.47 
16.42 
16.37 
16.35 

.990704 
.  990088 
.9900?:} 
.990057 
.990041 
.990025 
.990010 
.990594 

9.996578 
.996562 
.996546 
.996530 
.996514 
996498 
.996482 
.990405 
.990449 
.996433 

'  o~ 

/25 
.27 

jr 

.27 

i 

27 
.'27 
.28 
27 
.'27 
27 

.092266 
.098302 

.094386 
.095307 
.090395 
.097422 
.098446 
.099468 

|  9.100487 
I     .101504 
.102519 
.  103532 
.104542 
.  105550 
.106556 
.107559 
.108500 
.109559 

7.30 
7.27 
7.23 
7.18 
7.13 
7.12 
7.07 
7.03 
16.98 

16.95 
10.92 
16.88 
16.83 
16.80 
16.77 
16.72 
16.68 
16.65 
if?  f\9 

.908772 

.'.HKifiiW 

.905664 

.904688 

.903005 
.902578 
.901554 

.900532 

10.899513 
.898496 

.897481 
.896468 
.895458 
.894450 
.893444 
.892441 
.891440 
.890441 

58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.106973 
.  107951 
.  108927 
.  109901 
.110873 
.111842 
.112809 
.113774 
.114737 
.115698 

16.30 
16.27 
16.23 
16.20 
16.15  ; 
16.12 
16.08  ! 
16  05  ! 
16.02  i 
15.97 

9.996417 
.996400 
.996384 
.996368 
.990351 
.996335 
.996318 
.996302 
.996285 
.996269 

.28 
.27 
.27 

.28 

3 

.27 
.28 
.27 
28 

9.110556 
.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 
.118452 
.119429 

16.58 
16.53 
16.50 
10.47 
16.43 
16.40 
16.35 
16.33 
16.28 
16  25 

10.889444 
.888449 
.887457 
.886467 
.885479 
.884493 
.888509 
.882528 
.881548 
.880571 

39 
38 
37 
36 
35 
84 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9  116656 
.117613 
118567 
.119519 
.  120469 
.121417 
.  122302 
.123306 
.124248 
.125187 

15  95  i 
15.90  i 
15  87 
15.83 
15.80 
15.75 
15.73 
15.70 
15.05 
15  63 

9.996252 
.996235 
996219 
.996202 
.996185 
.996168 
.996151 
.990134 
.990117 
.996100 

.28 
.27 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 

9.120404 
.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 
.128130 
.129087 

16  22 
16.18 
16.15 
16.12 
16.08 
16.03 
16  02 
15.97 
15.95 
15  90 

10.879596 
.878623 
.877652 
.876683 
.875716 
.874751 
.873789 
.872828 
.871870 
.870913 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 

9  126125 
.127000 
.127993 
.128925 

15.58  > 
15.55 
15  53 
15  48 

9.996083 
.990066 
.996049 
.996032 

.28 
.28 
.28 

oo 

9.130041 
.130994 
.131944 
.132893 

15  88 
15.83 
15.82 

10.869959 
.869006 
.808056 
.867107 

19 
18 
17 
16 

45 
46 
47 
48 
49 
50 

129854 
130781 
131706 
132030 
133551 
134470 

15  45 
15.42  ! 
15.40 
15  35 
15.32 
15.28 

.996015 
.  995998 
.995980 
.  995963 
995946 
.995928 

.28 
'  .30 
.28 
28 
30 
.28 

133839 
.134784 
.  135726 
.136667 
.137605 
.138542 

15.75 
15.70 
15  68 
15  63 
15  62 
15  57 

.866161 
.865216 
.864274 
803333 
862395 
.861458 

15 
14 
13 
12 
11 
10 

51 
52 
53 

9  135387 
.136303 
137210 

15  27 

15  22 

3  995911 
995894 
.995876 

28 
.30 

9  139476 
.  140409 

.141340 

15.55 

15  52 

10  860524 
.859591 
866660 

9 

8 

7 

54 

138128  i 

.  995859 

28 

.142269 

15.48 

.857731 

6 

55 

139037 

995841 

143196 

S50N  >4 

5 

56 
57 
58 
59 
60 

139944 
140850 
.141754 
142055 
3  143555 

15  10    j 
15  07 
15  02 
15.00 

995823 
995806 
995788 
995771 
9  995753 

.28 
.30 
28 
.30 

.144121 
145044 
.145966 
146885 
9.147S03 

15  38 
15  37 
15.32 
15.30 

.  s:,.>79 
854956 

854084 
858115 

4 
3 
2 

0 

i 

Cosine. 

D.  r.   i 

Sine.     1 

D.  r.  i 

Cotang. 

D.  r. 

Tang. 

' 

97" 


183 


82" 


TABLE   XII. — LOGARITHMIC    SINES, 


171° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.143555 

9.995753 

qA 

9.147803 

1  fi  9^ 

10.852197 

60 

1 

2 
3 
4 
5 
6 

.144453 
.145349 
.146243 
.147136 
.148026 
.148915 

14.97 
14.93 
14.90 

14.88 
14.83 
14.82 

.995735 
.995717 
.995699 
.995681 
.995664 
.995646 

.OU 

.30 
.30 
.30 
.28 
.30 

OA 

.148718 
.  149632 
.150544 
.151454 
.152363 
.153269 

lO.^D 

15.23 
15.20 
15.17 
15.15 
15.10 

.851282 
.850368 
.849456 
.848546 
.847637 
.846731 

59 
58 
57 
56 
55 
54 

7 
8 
9 

.149802 
.150686 
.151569 

14.78 
14.73 

14  -I2 

.995628 
.995610 
.995591 

.OU 

.30 
.32 
30 

.154174 
.155077 
.155978 

15.08 
15.05 
15.02 

.845826 
.844923 
.844022 

53 
52 
51 

10 

.152451 

14.70 
14.65 

.995573 

.156877 

14^97 

.843123 

50 

11 
12 

9.153330 
.154208 

14.63 

9.995555 
.995537 

.30 

9.157775 

.158671 

14.93 

10.842225 
.841329 

49 

48 

13 
14 
15 
16 
17 
18 
19 

.155083 
.155957 
.156830 
.157700 
.158569 
.159435 
.160301 

14.58 
14.57 
14.55 
14.50 
14.48 
14.43 
14.43 

.995519 
.995501 
.995482 
.995464 
.995446 
.995427 
.995409 

.30 
.30 
.32 
.30 
.30 
.32 
.30 

qo 

.159565 
.160457 
.161347 
.162236 
.163123 
.164008 
.164892 

14.90 
14.87 
14.83 
14.82 
14.78 
14.75 
14.73 
14  70 

.840435 
.839543 
.838653 
.837764 
.836877 
.835992 
.835108 

47 
46 
45 
44 
43 
42 
41 

20 

.161164 

14.38 
14.35 

.995390 

.O.4 

.30 

.165774 

14^67 

.834226 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.162025 
.162885 
.  163743 
.164600 
.165454 
.166307 
.167159 
.168008 
.  168856 
.169702 

14.33 
14.30 
14.28- 
14.23 
14.24 
14.20 
14.15 
14.13 
14.10 
14.08 

9.995372 
.995353 
.995334 
.995316 
.995297 
.995278 
.995260 
.995241 
.995222 
.995203 

.32 
.32 
.30 
.32 
.32 
.30 
.32 
.32 
.32 
.32 

9.166654 
.167532 
.168409 
.169284 
.170157 
.171029 
.171899 
.172767 
.173634 
.174499 

14.63 
14.62 
14.58 
14.55 
14.53 
14.50 
14.47 
14.45 
14.42 
14.38 

10.833346 
.832468 
.831591 
.830716 
.829843 
.828971 
.828101 
.827233 
.826366 
.825501 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 

9.170547 

.171389 

14.03 

9.995184 
.995165 

.32 

9.175362 
.176224 

14.37 

10.824638 
.823776 

29 

28 

33 
34 
35 
36 
37 
38 
39 
40 

.172230 
.173070 
.173908 
.  174744 
.175578 
.176411 
.177242 
.178072 

14.02 
14.00 
13.97 
13.93 
13.90 
13.88 
13  85 
13.83 
13.80 

.  995146 
.995127 
.995108 
.995089 
.995070 
.995051 
.995032 
.995013 

.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.33 

.177084 
.177942 
.  178799 
.179655 
.180508 
.181360 
.182211 
.183059 

14.33 
14.30 
14.28 
14  27 
14.22 
14.20 
14.18 
14.13 
14.13 

.822916 
.822058 
.821201 
.820345 
.819492 
.818640 
.817789 
.816941 

27 
26 
25 
24 
23 
22 
21 
20 

41 
42 

9  178900 
.179726 

13.77 

9.994993 
.994974 

.32 
32 

9.183907 

.184752 

14.08 
14  08 

10.816093 
.815248 

19 
18 

43 
44 
45 
46 

47 

.180551 
.181374 
.182196 
.183016 
.183834 

13!  72 
13.70 
13.67 
13.63 

1  ^  P>9 

.994955 
.994935 
.994916 
.994896 
.994877 

!33 
.32 
.33 
.32 

qq 

.185597 
.186439 
.187280 
.188120 
.188958 

14.'  03 
14.02 
14.00 
13.97 

•jq  oq 

.814403 
.813561 
.812720 
.811880 
.811042 

17 
16 
15 
14 
13 

48 
49 

.184651 
.185466 

Jo.  \>6 

13.58 
13  57 

.994857 
.994838 

.00 

.32 
33 

.189794 
.190629 

Jo.yo 

13.92 

-«q  OQ 

.810206 
.809371 

12 
11 

50 

.186280 

13!  53 

.994818 

.191462 

JO.oo 

13.87 

.808538 

10 

51 
52 
53 
54 

9.187092 
.187903 
.188712 
.189519 

13.52 
13.48 
13.45 

9.994798 
.994779 
.994759 
.994739 

.32 
.33 
.33 

9.192294 
.193124 
.193953 
.1947'80 

13.83 

13.82 
13.78 

IQ  p"v 

10.807706 

.806876 
.806047 
.805220 

9 

8 
7 
6 

55 
56 
57 
58 
59 
60 

.190325 
.191130 
.191933 
.192734 
.193534 
9.194332 

13.43 
13.42 
13.38 
13.35 
13  33 
13  30 

.994720 
.994700 
.994680 
.994660 
.  994640 
9.994620 

.'33 
.33 
.33 
.33 
.33 

.195606 
.196430 
.197253 
.198074 
.198894 
9.199713 

Jo.  i  i 
13.73 
13.72 
13.68 
13.67 
13.65 

.804394 
.803570 
.802747 
.801926 
.801106 
10.800287 

5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang. 

' 

98« 


184 


81* 


COSINES,    TANGENTS,   AND   COTANGENTS. 


170° 


' 

Sine. 

D.r. 

Cosine. 

D.r. 

Tang. 

D.r. 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.194332 
.195129 
.195925 
.196719 
.197511 
.198302 
.199091 
.199879 
.200666 
.201451 
.202234 

. 

13.28 
13.27 
13.23 
13.20 
13.18 
13.15 
13.13 
13.12 
13.08 
13.05 
13.05 

9.994620 
.994000 
.994580 
.994500 
.994540 
.994519 
.994499 
.994479 
.994459 
.994438 
.994418 

.33 
.88 
.88 
.88 

.35 
.33 
.33 
.33 
.35 
..33 
.33 

!  9.199713 
.200529 
.201345 
.202159 
.202971 
.2037K2 
.204592 
.205400 
.206207 
.207013 
.207817 

13.60 
13.60 
13.57 
13.53 
13.52 
13.50 
13.47 
13.45 
13.43 
13.40 
13  37 

10.800287 
.799471 

.797841 

.796818 
.795408 
.794600 
.798798 
.798987 
.79X188 

60 
BO 

58 
57 
56 
55 

58 
51 

50 

11 
12 
18 
14 
15 
16 
17 
18 
19 
20 

9.203017 
.203797 
.204577 
.205354 
.206131 
.206906 
.207679 
.208452 
.209222 
.209992 

13.00 
13.00 
12.95 
12.95 
12.92 
12.88 
12.88 
12.83 
12.83 
12.80 

9.994398 
.994377 
.994357 
.994336 
.994316 
.994295 
.994274 
.994254 
.994233 
.994212 

.35 
.33 
.35 
.33 
.35 
.35 
.33 
.35 
.35 
.35 

9.208619 
.209420 
.210220 
.211018 
.211815 
.212611 
.213405 
.214198 
.214989 
.215780 

13.35 
13.33 
13.30 
13.28 
13.27 
13.23 
13.22 
13.18 
13.18 
13  13 

10.791381 
.790580 

.786595 
.785802 
.785011 

.784220 

40 
48 
47 
46 
45 
44 
.43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.210760 
.211526 
.212291 
.213055 
.213818 
.214579 
.215338 
.216097 
.216854 
.217609 

12.77 
12.75 
12.73 
12.72 
12.68 
12.65 
12.65 
12.62 
12.58 
12.57 

9.994191 
.994171 
.994150 
.994129 
.994108 
.994087 
.994066 
.994045 
.994024 
.994003 

.33 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

9.216568 
.217356 
.218142 
.218926 
.219710 
.220492 
.221272 
.222052 
.222830 
.223607 

13.13 
13.10 
13.07 
13.07 
13.03 
13.00 
13.00 
12.97 
12.95 
12.92 

10.783432 
.782644 
.781858 
.781074 
.780290 
.779508 
.778?28 
.777948 
.777170 
.770393 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 

9.218363 
.219116 
.219868 
.220618 
.221367 
.222115 
.222861 
.223606 

12.55 
12.53 
12.50 
12.48 
12.47 
12.43 
12.42 

19   QQ 

9.993982 
.993960 
.993939 
.993918 
.9938&T 
.993875 
.993854 
.993832 

.37 
.35 
.35 
.35 
.37 
.35 
.37 

9.224382 
.225156 
.225929 
.2267'00 
.227471 
.228239 
.229007 
.229773 

12.90 

12.88 
12.85 
12.85 
12.80 
12.80 
12.77 

10.775618 
.774844 
.774071 
.773300 
.778588 
.771701 
.770993 
770227 

29 
28 
27 
26 
85 
84 
88 
88 

39 
40 

.224349 
.225092 

12.38 
12.35 

.993811 
.993789 

.37 
.35 

.230539 
.231302 

12.77 
12.72 
12.72 

[769461 

.768698 

81 

20 

41 
42 
43 
44 

9.225833 
.226573 
.227311 

.228048 

12.33 
12.30 
12.28 

19  97 

9.993768 
.993746 
.993725 
.993703 

.37 
.35 
.37 

07 

9.232065 
.232826 
.233586 
.234345 

12.68 
12.67 
12.65 

19  A5t 

10.767935 

.707174 
.766414 

.70511:,* 

19 
18 
17 
16 

45 
46 
47 
48 
49 
50 

.228784 
.229518 
.230252 
.230984 
.231715 
.232444 

12.23 
12.23 
12.20 
12.18 
12.15 
12.13 

.993681 
.993660 
.993638 
.993616 
.993594 
.993572 

.35 
.37 
.37 
.37 
.37 
.37 

.235103 
.235859 
.236614 
.237368 
.238120 
.238872 

12.60 
12.58 
12.57 
12.53 
12.53 
12.50 

.764897 
.704141 
.788886 
.788688 

.761880 
.761128 

15 

14 

18 

12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
69 

9.233172 
.233899 
.234625 
.235349 
.236073 
.236795 
.237515 
.238235 
.238953 

12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 

9.993550 
.993528 
.993506 
.993484 
.993462 
.993440 
.993418 
.993396 
.993374 

.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 

9.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 
.244839 
.24557!) 

32.48 
12.45 
12.45 
12.42 
12.40 
12.38 
12.37 
12.33 

10.760378 
.759688 
.758888 
.758185 

9 
8 
7 
6 
5 
4 
8 
8 
1 

60 

9.239670 

11.95 

9.993351 

.38 

9.246319 

0 

' 

Cosine. 

D.r.  1 

Sine. 

D.r.  i! 

Cotang. 

D.r. 

Tang. 

' 

89° 


185 


80" 


10° 


TABLE   XII. — LOGARITHMIC    SINES, 


169° 


1 

Sine. 

D.  1". 

Cosine. 

D.I". 

Tang. 

D.I". 

Cotang. 

' 

0 

1 

9.239670 
.240386 

11.93 

9.993351 
.993329 

.37 

9.246319 
.247057 

12.30 

10.753681 
.752943 

60 

59 

2 

.241101 

11.92 

.993307 

.37 

.247794 

12.28 

.752206 

58 

3 
4 
5 

.241814 
.242526 
.24323? 

11.88 
11.87 
11.85 

noq 

.993284 
.993262 
.993240 

.38 
.37 
.37 

OQ 

.248530 
.249264 
.249998 

12.27 
12.23 
12.23 

.751470 
.750736 
.750002 

57 
56 
55 

6 

7 

.243947 
.244656 

.00 

11.82 

niro 

.993217 
.993195 

.OO 

.37 

OQ 

.250730 
.251461 

12.20 
12.18 

.749270 
.748539 

54 
53 

8 
9 
10 

.245363 
.246069 
.246775 

.  10 

11.77 
11.77 
11.72 

.993172 
.993149 
.993127 

.OO 

.38 
.37 

.38 

.252191 
.252920 
.253648 

12.17 
12.15 
12.13 
12.10 

.747809 
.747080 
.746352 

52 
51 
50 

11 
12 
13 
14 

9.247478 
.248181 

.248883 
.249583 

11.72 
11.70 
11.67 

9.993104 
.993081 
.993059 
.993036 

.38 
.37 
.38 

9.254374 
.255100 
.255824 
.256547 

12.10 
12.07 
12.05 

10.745626 
.744900 
.744176 
.743453 

49 

'48 
47 
46 

15 
16 

.250282 
.250980 

11.65 
11.63 

.993013 
.992990 

.38 
.38 

.257269 
.257990 

12.03 
12.02 

.742731 
.742010 

45 

44 

17 

.251677 

11.62 

.992967 

.38 

.258710 

12.00 

.741290 

43 

18 

.252373 

11  .60 

.992944 

.38 

.259429 

11  .98 

.740571 

42 

19 
20 

.253067 
.253761 

11.57 
11.57 
11.53 

.992921 
.992898 

.38 
.38 
.38 

.260146 
.260863 

11  .95 
11.95 
11.92 

.739854 
.739137 

41 
40 

21 

9.254453 

nKO 

9.992875 

9.261578 

10.738422 

39 

22 

.255144 

,ue 

.992852 

.38 

.262292 

11  .90 

.737708 

38 

23 

.255834 

11.50 

.992829 

.38 

.263005 

11.88 

.736995 

37 

24 

.256523 

11  .48 

.992806 

.38 

.263717 

11.87 

.736283 

36 

25 
26 

.257211 

.257898 

11.47 
11.45 

.992783 
.992759 

.38 
.40 

.264428 
.265138 

11.85 
11.83 

.735572 

.734862 

35 
34 

27 

.258583 

11.42 

.992736 

.38 

.265847 

11.82 

.734153 

33 

28 

.259268 

11  .42 

.992713 

.38 

.266555 

11.80 

.733445 

32 

29 

.259951 

11.38 

.992690 

.38 

.267261 

11.77 

.732739 

31 

30 

.260633 

11.37 
11.35 

.992666 

.40 
.38 

.267967 

11.77 
11.73 

.732033 

30 

31 

32 

9.261314 
.261994 

11.33 

9.992643 
.992619 

.40 

9.268671 
.269375 

11.73 

10.731329 

.730625 

29 

28 

33 
34 

.262673 
.263351 

11.32 
11.30 

HO7 

.992596 
.992572 

.38 
.40 

.270077 
.270779 

11.70 
11.70 

.729923 
.729221 

27 
26 

35 
36 
37 
38 
39 
40 

.264027 
.264703 
.265377 
.266051 
.266723 
.267395 

>iff 

11.27 
11.23 
11.23 
11.20 
11.20 
11.17 

.992549 
.992525 
.992501 
.992478 
.992454 
.992430 

.38 
.40 
.40 
.38 
.40 
.40 
.40 

.271479 
.272178 
.272876 
.273573 
.274269 
.274964 

11.67 
11.65 
11.63 
11.62 
11.60 
11.58 
11.57 

.728521 
.727822 
.727124 
.726427 
.725731 
.725036 

25 
24 
23 
22 
21 
20 

41 
42 

9.268065 
.2687:34 

11.15 

9.992406 

.992382 

.40 

9.275658 
.276351 

11.55 

10.724342 
.723649 

19 

18 

43 
44 

45 

.269402 
.270069 
.270735 

11.13 
11.12 
11.10 

nftS 

.992359 
.992335 
.992311 

.38 
.40 
.40 

.277043 
.277734 
.278424 

11.53 
11.52 
11.50 

.722957 
.722266 
.-721576 

17 
16 
15 

46 
47 
48 

.271400 
.272064 
.272726 

,\Jo 

11.07 
11.03 

.992287 
.992263 
.992239 

.40 
.40 
.40 

.279113 
.279801 
.280488 

11.48 
11.47 
11.45 

.720887 
.720199 
.719512 

14 

13 
12 

49 
50 

.273388 
.274049 

11.03 
11.02 
10.98 

.992214 
.992190 

.42 
.40 

.40 

.281174 
.281858 

11.43 
11.40 
11.40 

.718826 
.718142 

11 
10 

51 

9.274708 

1A  Qft 

9.992166 

9.282542 

10.717458 

9 

52 
53 

.275367 
.276025 

lU.yo 
10.97 

.992142 
.992118 

.40 
.40 

.283225 
.283907 

11.38 
11.37 

.716775 
.716093 

8 

7 

54 

.276681 

10.93 

.992093 

.42 

.284588 

11.35 

.715412 

6 

55 
56 

.277337 
.277991 

10.93 
10.90 

.992069 
.992044 

.40 
.42 

.285268 
.285947 

11.33 
11.32 

.714732 
.714053 

5 

4 

57 
58 

.278645 
.279297 

10.90 
10.87 

.992020 
.991996 

.40 
.40 

.286624 
.287301 

11.28 
11.28 

.713376 
.712699 

3 
2 

59 
.60 

.279948 
9.280599 

10.85 
10.85 

.991971 
9.991947 

.42  ' 
.40 

.287977 
9.288652 

11  .27 
11.25 

.712023 
10.711348 

1 
0 

' 

Cosine. 

D.I". 

Sine. 

D.  r. 

Cotang. 

D.  1".    Tang. 

' 

100° 


186 


79* 


11° 


COSINES,  TANGENTS;  AND  COTANGENTS.        lee" 


/ 

Sine. 

D.  1". 

Cosine. 

D.I". 

Tang. 

D.  r. 

Cotang. 

/ 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.280599 
.281248 
.281897 
.282544 
.283190 
.283836 
.284480 
.285124 
.285766 
.286408 
.287048 

10.82 
10.82 
10.78 
10.77 
10.77 
10.73 
10.73 
10.70 
10.70 
10.67 
10.67 

9.991947 
.991922 
.991897 
.991873 
.991848 
.991823 
.991799 
.991774 
.991749 
.991724 
.991699 

.42 
.42 
.40 
.42 
.42     ! 
.40 
.42 
.42 
.42 
.42 
.42 

9.288652 
.880606 

.289999 
.290671 
.291342 
.292013 
.292682 
.293350 
.294017 
.294684 
.295349 

11.23 
1  1  .  -,'-j 
11.20 
11.18 
11.18 
11.15 
11.13 
11.12 
11.12 
11.08 
11.07 

10.711348 
.710074 
.710001 
.709329 
.708658 
.707987 
.707318 
.706650 
.706868 
.705316 
.704651 

60 
68 

58 
67 

56 
55 
51 
53 
52 
51 
50 

11 
12 

9.287688 
.288326 

10.63 
in  fi^ 

9.991674 
.991649 

.42 

9.296013 
.296677 

11.07 

10.703987 
.703323 

49 

48 

13 

.288964 

1ft  Aft 

.991624 

.297339 

.702661 

47 

14 
15 
16 
17 
18 
19 

.289600 
.290236 
.290870 
.291504 
.292137 
.292768 

10.60 
10.57 
10.57 
10:55 
10.52 

1ft  ^9 

.991599 
.991574 
.991549 
.991524 
.991498 
.991473 

.42 
.42 
.42 
.43 
.42 

A9 

.298001 
.298662 
.299322 
.299980 
.300638 
.301295 

11.03 
11.02 
11.00 
10.97 
10.97 
10.95 

.701999 
.701338 
.700678 
.700020 
.699362 
.698705 

46 
45 
44 
43 
42 
41 

20 

.293399 

10.50 

.991448 

.43 

.301951 

10.93 

.698049 

40 

21 

9.294029 

1ft  A& 

9.991422 

A9 

9.302607 

10.697393 

39 

22 
23 
24 
25 
26 
27 
28 
29 
30 

.294658 
.295286 
.295913 
.296539 
.297164 
.297788 
.298412 
.299034 
.299655 

10.47 
10.45 
10.43 
10.42 
10.40 
10.40 
10.37 
10.35 
10.35 

.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 
.991193 

.42 
.43 
.42 
.43 
.42 
.43 
.43 
.42 
.43 

.303261 
.303914 
.304567 
.305218 
.305869 
.306519 
.307168 
.307816 
.308463 

10.88 
10.88 
10.85 
10.85 
10.83 
10.82 
10.80 
10.78 
10.77 

.696739 
.696086 
.695433 
.694782 
.694131 
.098481 
.692832 
.892184 
.691537 

38 
37 
36 
35 
34 
33 
32 
M 
30 

31 
32 
33 
34 
35 
36 
37 
38 

9.300276 
.300895 
.301514 
.302132 
.302748 
.303364 
.303979 
.304593 

10.32 
10.32 
10.30 
10.27 
10.27 
10.25 
10.23 

9.991167 
.991141 
.991115 
.991090 
.991064 
.991038 
.991012 
.990986 

.43 
.43 
.42 
.43 
.43 
.43 
.43 

9.309109 
.309754 
.310399 
.311042 
.311685 
.312327 
.312968 
.313608 

10.75 
10.75 
10.72 
10.72 
10.70 
10.68 
10.67 

10.690891 
.690246 
.689601 
.868968 

.688315 
.687673 
.687032 
.686392 

29 
28 
27 
26 
25 
24 
23 
22 

39 
40 

.305207 
.305819 

10.23 
10.20 
10.18 

.990960 
.990934 

.43 
.43 

.314247 
.314885 

10.63 
10.63 

.886758 

.685115 

21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.306430 
.307041 
.307650 
.308259 
.308867 
.309474 
.310080 
.310685 
.311289 
.311893 

10.18 
10.15 
10.15 
10.13 
10.12 
10.10 
10.08 
10.07 
10.07 
10  03 

9.990908 
.990882 
.990855 
.990829 
.990803 
.990777 
.990750 
.990724 
.990697 
.990671 

.43 
.45 
.43 
.43 
.43 
.45 
.43 
.45 
.43 
43 

9.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
319330 
.319961 
.320592 
.321222 

10.60 
10.60 
10.58 
10.57 
10.55 
10.55 
10.52 
10.52 
10.50 
10.48 

10.684477 
.683841 
.683205 
.882670 
.681936 
.681303 

.oswro 

.I5SM39 
JJT'.MOS 
.678778 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
65 
56 
57 
58 
59 
60 

9.312495 
.313097 
.313698 
.314297 
.314897 
.315495 
.316092 
.316689 
.317284 
9.317879 

10.03 
10.02 
9.98 
10.00 
9.97 
9.95 
9.95 
9.92 
9.92 

9.990645 
.990618 
.990591 
.990565 
.990538 
.990511 
.990485 
.990458 
.990431 
9.990404 

.45 
.45 
.43 
.45 
.45 
.43 
.45 
.45    i 
.45 

9.321851 
.322479 
.323106 
.323733 
.324358 
.3249a3 
.325607 
.326231 
.826858 
9.827175 

10.47 
10.45 
10.45 
10.42 
10.42 
10.40 
10.40 
10.37 
10.37 

10.678149 
.677521 
.678894 
.878887 

.CHOI;  10 

.675017 
.674393 

.878147 

9 
8 
7 
6 
5 
4 
3 
2 

0 

/ 

Cosine. 

D.I-. 

Sine. 

D.  I',    i 

Cotang.     D.  1'.  1 

Tang. 

/ 

101' 


187 


78* 


12° 


TABLE  XII. — LOGARITHMIC   SINES, 


167° 


' 

Sine/ 

D.  r. 

Cosine. 

D.r. 

Tang. 

D.l". 

Cotang. 

' 

0 

1 

9.317879 
.318473 

9.90 

9QO 

9.990404 
.990378 

.43 

9.327475 

.328095 

10.33 
in  v% 

10.672525 
.671905 

60 
59 

2 
3 

.-319066 
.319658 

.GO 

9.87 

9  OK 

.990351 
.990324 

!45 
45 

.328715 
.329334 

1U.OO 

10.32 
10  32 

.671285 
.670666 

58 
57 

4 
5 

.320249 

.320840 

.OO 

9.85 

.99029? 
.990270 

'.45 

.329953 
.330570 

10'.28 

.670047 
.669430 

56 
55 

6 

7 
8 
9 
10 

.321430 
.322019 
.322607 
.323194 
.323780 

9.83 
9.82 
9.80 
9.78 
9.77 
9.77 

.990243 
.990215 
.990188 
.990161 
.990134 

.45 
.47 
.45 
.45 
.45 
.45 

.331187 
.331803 
.332418 
.333033 
.333646 

10.28 
10.27 
10.25 
10.25 
10.22 
10.22 

.668813 
.668197 
.667582 
.666967 
.666354 

54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 

9.324366 
.324950 
.325534 
.326117 
.326700 
.327281 

9.73 
9.73 
9.72 
9.72 

9.68 
9fift 

9.990107 
.990079 
.990052 
.990025 
.989997 
.989970 

.47 
.45 
.45 
.47 
.45 

9.334259 
.334871 
.335482 
.336093 
.336702 
.337311 

10.20 
10.18 
10.18 
10.15 
10.15 
in  1  Q 

10.665741 
.665129 
.664518 
.663907 
.663298 
.662689 

49 
48 
47 
46 
45 
44 

17 

18 
19 
20 

.327862 
.328442 
.329021 
.329599 

.Do 

9.67 
9.65 
9.63 
9.62 

.989942 
.989915 
.989887 
.989860 

'.45 

.47 
.45 
.47 

.337919 
.338527 
.339133 
.339739 

lU.  lo 
10.13 
10.10 
10.10 
10.08 

.662081 
.661473 
.660867 
.660261 

43 
42 
41 

40 

21 
22 
23 
24 

9.330176 
.330753 
.331329 
.331903 

9.62 
9.60 
9.57 

9.989832 
.989804 
.989777 
.989749 

.47 
.45 
.47 

9.340344 
.340948 
.341552 
.342155 

10.07 
10.07 
10.05 

10.659656 
.659052 
.658448 
.657845 

39 
38 
37 
36 

25 
26 

27 
28 
29 
30 

.332478 
.333051 
.333624 
.334195 
.334767 
.335337 

9.58 
9.55 
9.55 
9.52 
9.53 
9.50 
9.48 

.989721 
.989693 
.989665 
.989637 
.989610 
.989582 

.47 
.47 
.47 
.47 
.45 
.47 
.48 

.342757 
.343358 
I  .343958 
.344558 
.345157 
.345755 

10.03 
10.02 
10.00 
10.00 
9.98 
9.97 
9.97 

.657243 
.656642 
.656042 
.655442 
.654843 
.654245 

35 
34 

32 
31 
30 

31 

9.335906 

9.989553 

9.346353 

10.653647 

29 

32 

.336475 

•*•  ^ 

.989525 

.47 

.346949 

9QQ 

.653051 

28 

33 
34 
35 
36 

.337043 
.337610 
.338176 
.338742 

9.47 
9.45 
9.43 
9.43 

Q  49 

.989497 
.989469 
.989441 
.989413 

.47 
.47 
.47 
.47 

.347545 
.348141 
.348735 
.349329 

.  yo 
9.93 
9.90 
9.90 
9  88 

.652455 
.651859 
.651265 
.650671 

27 
26 
25 
24 

37 

.339307 

y  .e±& 

Q  4fi 

.989385 

*4ft 

.349922 

.650078 

23 

38 

.339871 

y  .4u 

Q  ^8 

.989356 

,4o 

.350514 

9  87 

.649486 

22 

39 
40 

.340434 
.340996 

y  .00 
9.37 
9.37 

.989328 
.989300 

!47 
.48 

.351106 
.351697 

9^85 
9.83 

.648894 
.648303 

21 
20 

41 

42 

9.341558 
.342119 

9.35 

Q  VI 

9.989271 
.989243 

.47 

9.352287 
.352876 

9.82 

10.647713 
.647124 

19 
18 

43 

.342679 

y.oo 
900 

.989214 

Art 

.353465 

9  DA 

.646535 

17 

44 
45 
46 

.343239 
.343797 
.344355 

.00 
9.30 
9.30 

Q  9ft 

.989186 
.989157 
.989128 

.47 

.48 
.48 

.354053 
.354640 
.355227 

.oU 

9.78 
9.78 

9r»r> 

.645947 
.645360 
.644773 

16 
15 
14 

47 

.344912 

y  .Aio 

.989100 

.47 

.355813 

.  <  < 

9r<(r 

.644187 

13 

48 

.345469 

0  OK 

.989071 

.48 

.356398 

.  <  O 

.643602 

12 

49 

.346024 

9.XO 

n  OK 

.989042 

"^ 

.356982 

9»~o 

.643018 

11 

50 

.346579 

9^25 

.989014 

.47 

.48 

.357566 

.  <o 

9.72 

.642434 

10 

51 

52 

9.347134 

.347687 

9.22 

999 

9.988985 
.988956 

.48 

9.358149 
.358731 

9.70 

q  I~A 

10.641851 
.641269 

9 

S 

53   .348240 

.MB 

9  20 

.988927 

4ft 

.359313 

9'fff 

.640687 

7 

54   .348792 

91ft 

.988898 

.4o 
4ft 

.359893 

.D< 
9fift 

.640107 

6 

55 
56 
57 
58 
59 

.349343 
.349893 
.350443 
.350992 
.351540 

.  15 

9.17 
9.17 
9,15 
9.13 

n  1  o 

.988869 
.988840 
.988811 
.988782 
.988753 

.4o 

.48 
.48 
.48 
.48 

/4ft 

.360474 
.361053 
.361632 
.362210 

.362787 

.uo 
9.65 
9.65 
9.63 
9.62 

.639526 
.638947 
.638368 
.637790 
.637213 

5 
4 
3 
2 
1 

60  9.352088   ">10 

9.988724 

.4o 

9.363364 

9.62 

10.636636 

0 

'  I  Cosine.   D.  1".  |l  Sine,  |  D.  1". 

Cotang. 

D.  1*.   Tang. 

' 

188 


77* 


COSINES,    TANGENTS,    AND   COTANGENTS.  166° 


' 

Sine. 

D.I'. 

Cosine. 

D.  r.  |  Tang. 

D.  r. 

Cotang. 

• 

0  9.352088 
1   .352635 

9.12 
9  10 

9.988724 
.988695 

,o    9.363364 
•Jjj     .363040 

9.60 

10.636036 
.030000 

60 
59 

2 
3 

.853181 
.353726 

9[  08 

.988666 

.988036 

!  365090 

9.56 

9.58 

.035485 

57 

4 

354271    '  v'- 

.988607 

4s   -;^5C6i 

9.57 

.034336 

56 

5 

'.354815    o'ar 

.988578 

9  .  55 

55 

6 

.355358   V  "V! 

.988548 

!  866810 

9.66 

.688190 

54 

7 

.355901 

j  .  \-><) 

.988519 

•?;:     .367382 

9.53 

.032018 

58 

8 

.356443 

o  no 

.988489 

.867968 

9.52 

.032047 

52 

9 

.356984 

0  00 

.988460 

•g  :  .368524 

!'  .  52 

.031476 

51 

10 

.357524 

9[oo 

.988430 

•Jg   ;  .309094 

9.48 

.630906 

50 

11 

9.358064 

8  no 

9.988401 

',n   9.309663 

10.630337 

49 

12 

.358603 

.yo 

.988371 

.370232 

9.48 

.629768 

48 

13 
14 
15 

.359141 
.359678 
.360215 

8.97 
8.95 
8.95 

8Qt 

.988:342 
.988312 

.988282 

•Jg  i]  .370799 
•JX    -371367 

.371  1(33 

9.45 
9.47 
9.43 

.029201 
.628688 
.688067 

47 
46 
45 

16 

.360752 

.yo 

8  GO 

;  988252 

•JH   ;  .372499 

9.43 

.627501 

44 

17 
18 
19 
20 

.361287 
.361822 
.362356 
.362889 

.We 

8.92 
8.90 

8.88 
8.88 

.988223 
.988193 
.988163 
.988133 

•J2   i  -373064 
•2J     .373629 
i  .374193 
•;*;   i  .374756 

9  '.42 
9.40 
9.38 
9.38 

.626936 
.620371 
.625807 
.625244 

43 
42 
41 

40 

21 
22 
"23 
24 

9.363422 
.363954 
.364485 
365016 

8.87 
8.85 
8.85 

9.988103 
.988073 
.988043 
.988013 

'"    9.375319 
.375881 
•rj   ,   -376442 
'I.,     .377003 

9.37 
9.  '35 
9.35 

900 

10.624681 
.024119 
.623558 
.622997 

39 
38 
87 
36 

26 
27 
28 
29 
30 

.365546 
.366075 
.366604 
.367131 
.367659 
.368185 

8.83 
8.82 
8.82 
8.78 
8.80 
8.77 
8.77 

.9879813 
.987953 
1  .987922 
.987892 
.987862 
.987832 

[50 
.52 
.50 
.50 
.50 
.52 

.377563 
.378122 
.378681 
.379239 
.379797 
.380354 

.Oo 

9.32 
9.32 
9.30 
9.30 
9.28 
9  27 

.622437 
.621878 
.621319 
.620761 
.620203 
.619646 

35 
84 

33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.368711 
.369236 
.369761 
.370285 
.370808 
.371,330 
.371852 
.372373 
.372894 
.373414 

8.75 
8.75 
8.72 
8.72 
8.70 
8.70 
8.68 
8.68 
8.67 

8C.fi 

9.987801 
1  .987771 
.987740 
.987710 
.987679 
.987649 
.987618 
.987588 
.987557 
.987526 

.50 
.52 
.50 
.52 

.50 
.52 
.50 
.52 
.52 
.50 

9.380910 
.381466 
.382020 
.382575 
.383129 
.383682 
.384234 
.384786 
.385337 
.385888 

9.27 
9.23 
9.25 
9.23 
9.22 
9.20 
9.20 
9.18 
9.18 
9.17 

10.619090 
.618534 
.617980 
.617425 
.616871 
.616318 
.015700 
.615214 
.614663 
.614112 

29 
28 
27 

25 
24 

a 

22 

n 

20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.373933 
.374452 
.374970 
.375487 
.376003 
.376519 
.3770,35 
.377549 
.378063 
.378577 

9-379089 
.379601 
.380113 
.380624 
.381134 
.381643 
.382152 
.382661 
.383168 
9.383675 

.DO 

8.65 
8.63 
8.62 
8.60 
8.60 
8.60 
8.57 
8.57 
8.57 
8.53 

8.53 
8.53 
8.52 
8.50 
8.48 
8.48 
8.48 
8.45 
8.45 

9.987496 
.987465 
.987434 
.987403 
.987372 
.987341 
.987310 
.987279 
.987248 
.987217 

9.987186 
.987155 
.987124 
.987092 
.987061 
.987030 
.986998 
.986967 
.986936 
9.986904 

.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 

.52 
.52 
.53 
.52 
.52 
.53 
.52 
.52 
.53 

9.386438 
.386987 
.387536 
.388084 
I  .388631 
.389178 
!  .389724 
I  .390270 
.390815 
.391360 

9.391903 
.392447 
.302989 
.31*3531 
.394073 
.394614 
.395154 
.895694 
.896388 
9.396771 

9.15 
9.15 
9.13 
9.12 
9.12 
9.10 
9.10 
9.0f 
9.08 
9.05 

9.07 
9.03 
9.03 
9.03 
9.01 
9.00 
9.00 
8.0 
8.97 

10.613562 
.613013 
.612464 
.611916 
.611869 
.610822 
.610270 

.609186 

.608640 

10.608087 
.607558 
.607011 
.606469 
.605927 
.606186 
[604840 
[604806 
.608707 
10.603229 

19 
18 
17 
16 
15 
14 
13 

11 
10 

9 

8 
7 
6 

I 

8 

1 

0 

. 

/ 

Cosine. 

D.  r. 

Sine. 

D.  1".   Cotang. 

D.  r. 

Tang.  I  ' 

1  7.6' 

103' 


189 


14° 


TABLE  XII. — LOGARITHMIC   SINES, 


165° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.383675 

8AK. 

9.986904 

52 

9.396771 

8Q7 

10.603229 

60 

1 

.384182 

.10 
8  An 

.986873 

.397309 

.y< 

8  QK 

.602691 

59 

2 
3 

4 

.384687 
.385192 
.385697 

,*Bi 

8.42 
8.42 

.986841 
.986809 
.986778 

!53 
.52 

KO 

.397846 
.398383 
.398919 

.yo 

8.95 
8.93 

8  QO 

.602154 
.601617 
.601081 

58 
57 
56 

5 
6 

.386201 
.386704 

8^38 

8OQ 

.986746 
.986714 

.Do 

.53 

.399455 
.399990 

.  yo 
Son 

.600545 
.600010 

55 

54 

7 
8 

.387207 
.387709 

.  OO 

8.37 

8  OK 

.986683 
.986651 

!53 

KO 

.400524 
.401C58 

.yu 
8.90 

8QO 

.599476 
.598942 

53 

52 

9 

.388210 

.oD 

.986619 

.Do 

KO 

.401591 

.OO 
800 

.598409 

51 

10 

.388711 

8.35 
8.33 

.986587 

.Do 

.53 

.402124 

.OO 

8.87 

.597876 

50 

11 

9.389211 

800 

9.986555 

KO 

9.402656 

8QK 

10.597344 

49 

12 
13 

.389711 
.390210 

.  OO 

8.32 

8  on 

.986523 
.986491 

.Do 

.53 

KO 

.403187 
.403718 

.OD 

8.85 
8  85 

.596813 
.596282 

48 
47 

14 
15 

.390708 
.391206 

.ou 

8.30 

.986459 
.986427 

.Oo 

.53 

.404249 
.404778 

8.  '82 

8QO 

.595751 
.595222 

46 
45 

16 

.391703 

8.28 

897 

.986395 

.53 

to 

.405308 

.OO 
8  Oft 

.594692 

44 

17 
18 
19 
20 

.392199 
.392695 
.393191 
.393685 

.4( 

8.27 
8.27 
8.23 
8.23 

.986363 
.986331 
.986299 
.986266 

.Do 

.53 
.53 
.55 
.53 

.405836 
.406364 
.406892 
.407419 

.  oil 

8.80 
8.80 
8.78 
8.77 

.594164 
.593636 
.593108 
.592581 

43 
42 
41 
40 

21 

9.394179 

9.986234 

KO 

9.407945 

877 

10.592055 

39 

22 
23 

.394673 
.395166 

8^22 

.986202 
.986169 

.Do 

.55 

pro 

.408471 
.408996 

.  <  < 

8.75 
8  75 

.591529 
.591004 

38 
37 

24 
25 

26 

.395658 
.396150 
.396641 

8^20 
8.18 

.986137 
.986104 
.986072 

.Do 

.55 
.53 

.409521 
.410045 
.410569 

8.73 
8.73 

ft  "~O 

.590479 
.589955 
.589431 

36 
35 
34 

27 
28 

.397132 
.397621 

8.18 
8.15 
81" 

.986039 
.986007 

.55 
.53 

.411092 
.411615 

8  '.72 
8  70 

.588908 
.588385 

33 
32 

29 

.398111 

.14 

.985974 

.55 

.412137 

8  Aft 

.587863 

31 

30 

.398600 

8.15 
8.13 

.985942 

.53 
.55 

.412658 

.  Do 

8.68 

.587342 

30 

31 
32 

9.399088 
.399575 

8.12 

9.985909 
.985876 

.55 

9.413179 
.413699 

8.67 

10.586821 
.586301 

29 

28 

33 

.400062 

8.12 

.985843 

.55 

KO 

.414219 

8f>K 

.585781 

27 

34 

.400549 

8.  12 
8m 

.985811 

.Do 

.414738 

.  OO 
8  OK 

.585262 

26 

35 

.401035 

.10 
8  Aft 

.985778 

.55 

.415257 

.  DD 

8(\r> 

.584743 

25 

36 
37 

38 

.401520 
.402005 
.402489 

.Uo 
8.08 
8.07 

.985745 
.985712 
.985679 

.55 
.55 
.55 

.415775 
.416293 
.416810 

.Do 

8.63 
8.62 

80ft 

.584225 
.583707 
.583190 

24 
23 
22 

39 
40 

.402972 
.403455 

8.'  05 
8.05 

.985646 
.985613 

.55 
.55 
.55 

.417326 
.417842 

.  DU 

8.60 
8.60 

.582674 
.582158 

21 
20 

41 

42 
43 

9.403938 
.404420 
.404901 

8.03 
8.02 

9.985580 
.985547 
.985514 

.55 
.55 

9.418358 

.418873 
.419387 

8.58 
8.57 

8K7 

10.581642 
.581127 
.580613 

19 

18 
17 

44 
45 

.405382 
.405862 

8.02 
8.00 

rj  QQ 

.985480 
.985447 

!55 

KK 

.419901 
.420415 

.04 

8.57 

Q  KK 

.580099 
.579585 

16 
15 

46 
47 
48 
49 
50 

.406341 
.406820 
.407299 
.407777 
.408254 

<  .yo 
7.98 
7.98 
7.97 
7.95 
7.95 

.985414 
.985381 
.985347 
.985314 
.985280 

.DO 

.55 
.57 
.55 
.57 
.55 

.420927 
.421440 
.421952 
.422463 
.422974 

o.DD 

8.55 
8.53 
8.52 
8.52 
8.50 

.579073 
.578560 
.578048 
.577537 
.577026 

14 
13 
12 
11 
10 

51 

9.408731 

7QQ 

9.985247 

9.423484 

8AQ 

10.576516 

9 

52 

.409207 

.yo 

.985213 

*** 

.423993 

.4O 

8Kfl 

.576007 

8 

53 
54 

.409682 
.410157 

7.93 
7.92 

.985180 
.985146 

.55 
.57 

.424503 
.425011 

.  DU 

8.47 

8  Art 

.575497 
.574989 

7 
6 

55 

.410632 

7.92 

.985113 

.55 

.425519 

.44 

847 

.574481 

5 

56 

.411106 

7.90 

.985079 

*£» 

.426027 

.4< 

8AK 

.573973 

4 

57 

.411579 

7.88 

7QQ 

.985045 

.5< 

K7 

.426534 

.4D 

.573466 

3 

58 

.412052 

.00 

.985011 

.D< 

.427041 

j>"  j2 

.572959 

2 

59 
60 

.412524 
9.412996 

7.87 
7.87 

.984978 
9.984944 

.55 
.57 

.427547 
9.428052 

8.'43 

.572453 
10.571948 

1 
0 

'fl 

Cosine. 

D.  r. 

Sine. 

D.  1'. 

Cotang. 

D.  r. 

Tang,  j 

' 

104° 


190 


75° 


CQSINES,    TANGENTS,   AND    COTANGENTS.  164° 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1'. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 

3 
4 
5 
6 

7 
8 
9 
10 

9.412996 
.413467 
.413938 
.414408 
.414878 
.415347 
.415815 
.416283 
.416751 
.417217 
.417684 

7.85 

7.85 
7.83 
7.83 
7.82 
7.80 
7.80 
7.80 
7.77 

7^77 

9.984944 
.984910 
.984876 
.984842 
.984808 
.984774 
.984740 
.984706 
.984672 
.984638 
.984603 

.57 
.57 
.57 
.57 
.57 
.57 
.57 
.57 
.57 
.58 
.57 

9.428052 
.428558 
.429062 
.429566 
.430070 
.430573 
.431075 
.431577 
.432079 
.432580 
.433080 

8.43 
8.40 
8.40 
8.40 
8.38 
8.37 
8.37 
8.37 
8.35 
8.33 
8.33 

10.571948 
.571442 
.570938 
.570434 
.569930 
.569427 
.668085 
.568423 
.507981 
.567420 
.566920 

60 
59 
58 
57 
56 
55 
54 
53 
B8 
51 
50 

11 

9.418150 

7  75 

9.984569 

57 

9.433580 

Q  OQ 

10.566420 

49 

12 

.418615 

.984535 

.434080 

O.oO 

.565920 

48 

13 
14 
15 
16 
17 
18 
19 
20 

.419079 
.419544 
.420007 
.420470 
.420933 
.421395 
.421857 
.422318 

7!  75 

7.72 
7.72 
7.72 
7.70 
7.70 
7.68 
7.67 

.984500 
.984466 
.984432 
.984397 
.984363 
.984328 
.984294 
.984259 

!57 
.57 
.58 
.57 
.58 
.57 
.58 
.58 

.434579 
.435078 
.435576 
.436073 
.436570 
.437067 
.437563 
.438059 

8.32 
8.32 
8.30 
8.28 
8.28 
8.28 
8.27 
8.27 
8.25 

.565421 
.564922 
.564424 
.563927 
.563430 
.562933 
.562437 
.561941 

47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 

9.422778 
.423238 
.423697 
.424156 

7.67 
7.65 
7.65 

9.984224 
.984190 
.984155 
.984120 

.57 
.58 
.58 

to 

9.438554 
.439048 
.439543 
.440036 

8.2a 
8.25 
8.22 

899 

10.561446 
.560952 
.560457 
.559964 

39 
38 
37 
36 

25 
26 

.424615 
.425073 

7.65 
7.63 

.984085 
.984050 

.DO 

.58 

.440529 
.441022 

.£& 

8.22 

.559471 
.558978 

35 
34 

27 

.425530 

7.62 

.984015 

.58 

.441514 

8.20 

.558486 

33 

28 

425987 

7.62 

.983981 

.57 

.442006 

8.20 

.557994 

32 

29 

.426443 

7.60 

.983946 

.58 

fro 

.442497 

8.18 

81B 

.557503 

31 

30 

.426899 

7.60 
7.58 

.983911 

.Do 

.60 

.442988 

.  io 
8.18 

.557012 

30 

31 

9.427354 

9.983875 

to 

9.443479 

81K 

10.556521 

29 

32 

.427809 

7.58 

.983840 

.Do 
to 

.443968 

.  ID 
817 

.556032 

28 

33 

.428263 

7.57 

.983805 

.DO 

tQ 

.444458 

.if 

81^ 

.555542 

27 

34 

.428717 

7.57 

.983770 

.Do 
KB 

.444947 

.  ID 

.555053 

26 

35 

.429170 

7.55 

.983735 

.DO 

.445435 

8.13 

.354565 

25 

36 

.429623 

7.55 

.983700 

.58 
fift 

.445923 

8.13 

8-jq 

.554077 

24 

37 
38 
39 
40 

.430075 
.430527 
.430978 
.431429 

7.53 
7  53 
7.52 
7.52 
7.50 

.983664 
.983629 
.983594 
.983558 

.bU 

.58 
.58 
.60 
.58 

.446411 

.446898 
.447384 
.447870 

.  lo 

8.12 
8.10 
8.10 
8.10 

.553589 
.553102 
.552616 
.552130 

23 
22 
21 
20 

41 

9.431879 

9.983523 

ftn 

9.448356 

81  K 

10.551644 

19 

42 
43 
44 
45 
46 
47 
48 
49 
50 

.432329 
.432778 
.4a3226 
.433675 
.434122 
.434569 
.435016 
.435462 
.435908 

7.50 
7.48 
7.47 
7.48 
7.45 
7.45 
7.45 
7.43 
7.43 
7.42 

.983487 
.983452 
.983416 
.983381 
.983345 
.983309 
.983273 
.983238 
.983202 

.bU 
.58 
.60 
.58 
.60 
.60 
.60 
.58 
.60 
.60 

.448841 
.449326 
.449810 
.450294 
.450777 
.451260 
.451743 
.452225 
.452706 

.Uo 

8.08 
8.07 
8.07 
8.05 
8.05 
8.05 
8.03 
8.02 
8.02 

.551159 
.550674 
.550190 
.549706 
.540223 
.548740 
.548257 
.547775 
.547294 

18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.436353 
.436798 
.437242 
.437686 
.438129 
.438572 
.439014 
.439456 
.439897 
9.440338 

7.42 
7.40 
7.40 
7.38 
7.38 
7.37 
7.37 
7.35 
7.35 

9.983166 
.983130 
.983094 
.983058 
.983022 
.982986 
.982950 
.982914 
.982878 
9.982842 

.60 
.60 
.60 
.60 
.60 
.  .60 
.60 
.60 
.60 

9.453187 
.453668 
.454148 
.454628 
.455107 
.455586 
.456064 
.456542 
.457019 
9.457496 

8.02 
8.00 
8.00 
7.98 
7.98 
7.97 
7.97 
7.95 
7.95 

10.546813 
.546332 
.M.J.SM 

!544898 

.544414 
.543936 

.5-i.ms 
.548981 
10.542504 

9 
8 
7 
6 
5 
4 

2 
1 
0 

*  \  Cosine. 

D.  I'. 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

105- 


191 


16° 


TABLE   XII. — LOGARITHMIC    SINES, 


163° 


' 

Sine. 

D.I". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.440338 
.440778 

7.33 

7  33 

9.982842 
.982805 

.62 

9.457496 
.457973 

7.95 

10.542504 
.542027 

60 
59 

2 
3 
4 

.441218 
.441658 
.442096 

7^33 
7.30 
7  32 

.982769 
.982733 
.982696 

'.60 
.62 

.458449 
.458925 
.459400 

7.93 
7.93 
7.92 

.541551  58 
.541075  57 
.540600  56 

5 
6 

7 
8 

.442535 
.442973 
.443410 
.443847 

7^30 
7.28 
7.28 
7  28 

>!  .982660 
.982624 
.982587 
.982551 

!eo 

.62 
.60 

.459875 
.460349 
.460823 
.461297 

7.92 
7.90 
7.90 
7.90 

r-  on 

.540125  55 
.539651   54 
.539177  i  53 
.538703  52 

9 

.444284 

.982514 

*RO 

.461770 

<  .OO 

.538230  51 

10 

.444720 

7^25 

.982477 

'.60 

.462242 

7.87 
7.88 

.537758  50 

11 

9.445155 

7  25 

9.982441 

9.462715 

10.537285  49 

12 

.445590 

7*25 

.982404 

Rf> 

.463186 

Z'52 

.536814  48 

13 

.446025 

7  23 

.982367 

.uSB 

60 

.463658 

1  .O< 
r-  oq 

.536342  47 

14 

.446459 

7*2,3 

.982331 

62 

.464128 

t  .OO 

.535872  46 

15 

.446893 

7  22 

.982294 

62 

.464599 

ii*S| 

.535401  45 

16 
17 

.447326 
.447759 

l'-22 

.982257 
.982220 

.*62 

..465069 
.465539 

7^83 

.534931  44 
.534461  43 

18 

.448191 

7*90 

.982183 

RO 

.466008 

7.82 

7  BO 

.533992 

42 

19 
20 

.448623 
.449054 

riia 

7.18 

.982146 
.982109 

!62 
.62 

.466477 
.466945 

t  .0/4 

7.80 
7.80 

.533523 
.533055 

41 
40 

21 

9.449485 

7  17 

9.982072 

9.467413 

r»  r>Q 

10.532587 

39 

22 

.449915 

.982035 

ro 

.467880 

t  .  t  0 

f  r-o 

.532120 

38 

23 

.450345 

717 

.981998 

RO 

.468347 

I  .  <O 
r-  r-o 

.531653 

37 

24 

.450775 

7  15 

.981961 

CO 

.468814 

i  .  tO 

.531186 

36 

25 

.451204 

7  13 

.981924 

.  D/4 

CO 

.469280 

7.77 

.530720 

35 

26 

.451632 

i  .  13 

.981886 

.DO 

.469746 

7.77 

.530254 

34 

27 

.452060 

7  13 

.981849 

.  62 

RO 

.470211 

7.75 

.529789 

33 

28 

.452488 

7  12 

.981812 

.O/6 
co 

.470676 

7.  <5 

.529324 

32 

29 

.452915 

.981774 

.Do 

RO 

.471141 

7.75 

r>  i~o 

.528859 

31 

30 

.453342 

7.'  10 

.981737 

.D^ 

.62 

.471605 

i  .to 

7.73 

.528395 

30 

31 

9.453768 

7  10 

9.981700 

RP. 

9.472069 

10.527931 

29 

32 

.454194 

r-'no 

.981662 

.Do 

.472532 

Z*i? 

527468 

28 

33 

.454619 

i  .Uo 

7  08 

.981625 

.62 

!  .472995 

»  .  fit 

.527005  !  27 

34 

.455044 

.981587 

CO 

.473457 

(t'r-r! 

.526543  26 

35 

36 

.455469 
.455893 

7\  07 

.981549 
.981512 

.DO 

.62 

.473919 
.474381 

7.  iO 

7.70 

7  Rft 

.526081  25 
.525619  24 

37 

.456316 

7  05 

.981474 

.63 

RP 

.474842 

i  .Do 

.525158  '  23 

38 
39 

'.456739 
.457162 

7^05 
7  03 

.981436 
.981399 

.Do 

.62 

RP. 

.475303 
.4T57G3 

7.67 

.524697  ,  22 
.524237  i  21 

40 

.457584 

.981361 

.DO 

.63 

.476223 

7.  '67 

.523777 

20 

41 

9.458006 

7  02 

9.981323 

9.476683 

r<  R^ 

10.523317 

19 

42 

.458427 

.981285 

CO 

.477142 

t  .DO 

*"  R^ 

.522858 

18 

43 

.458848 

7  00 

.981247 

.DO 

CO 

.477601 

i  .DO 

.522399 

17 

44 

.459268 

.981209 

.DO 

.478059 

£*S 

.521941 

16 

45 

.459688 

7nn 

.981171 

no 

.47'8517 

Z*2 

.521483 

15 

46 

.460108 

.uu 
6  98 

.981133 

.DO 
RP. 

.478975 

<  .DO 

.521025 

14 

47 

.460527 

6  no 

.981095 

.Do 

.479432 

'Z*S 

.  520568* 

13 

48 

.460946 

.yo 

6Q7 

.981057 

.63 

.479889 

<  .o2 

.520111 

12 

49 

50 

.461364 
.461782 

.  y< 
6.97 
6.95 

.981019 
.980981 

!63 
.65 

.480345 
.480801 

7\60 
7.60 

.519655 
.519199 

11 
10 

51 

9.462199 

6  95 

9.980942 

cq 

9.481257 

*   KQ 

10.518743 

9 

52 

.462616 

6oq 

.980904 

.  Do 

.481712 

7  ^ft 

.518288 

8 

53 

54 

.463032 
.463448 

.  yo 
6.93 

.980866 
.980827 

!65 

.482167 
.482621 

7^57 

.517833 
.517379 

6 

55 

.463864 

6.93 

.980789 

.63 

.483075 

7.57 

.516925 

5 

56 

.464279 

6.  92 

.980750 

.65 

.483529 

7.57 

.516471 

4 

57 

.464694 

6.92 
6  90 

.980712 

.63 
65 

.4&S982 

7.55 

.516018 

3 

58 

.465108 

6  on 

.980673 

.484435 

7  ^P 

.515565 

2 

59 

.465522 

.  yu 

600 

.980635 

'<£ 

.484887 

r*  KO 

.515113 

1 

60 

9.465935 

.00 

9.980596 

.DO 

9.485339 

(  .OO 

10.514661 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

100* 


192 


73* 


17° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


162° 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

3 
4 
5 
6 
7 
8 
9 
10 

9.485935 
.466348 
.466761 
.467173 
.467585 
.467996 
.468407 
.468817 
.469227 
.469637 
.470046 

6.88 
6.88 
6.87 
6.87 
6.85 
6.85 
6.83 
6.83 
6.83 
6.82 
6.82 

i  9.980596 
.980558 
.980519 
.980480 
.980442 
.980403 
.980364 
.980325 
.980286 
.980247 
.980208 

.63 

.05 

!<J3 
.65 

.05 
.05 
.05 
.65 
.65 
.05 

9.485339 
.485791 

.486242 
.486693 
.487143 
.487593 
.488043 
.488492 
.488941 
.489390 
.489838 

7.53 

7.58 

7.50 
7.50 
7.50 
7.48 
7.48 
7.48 
7.47 
7.47 

10.514661 
.514209 

.513307 
.518857 

.511957 
.511508 
.511059 
.510010 
.510162 

60 
59 
58 
57 
56 
55 
54 
56 
52 
51 
50 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 

9.470455 
.470863 
.471271 
.471679 
.472086 
.472492 
.472898 
.473304 
.473710 
.474115 

6.80 
6.80 
6.80 
6.78 
6.77 
6.77 
6.77 
6.77 
6.75 
6.73 

9.980169 
.980130 
.980091 
.980052 
.980012 
.979973 
.979934 
.979895 
.979855 
.979816 

.65 
.65 
.65 
.67 
.65 
.05 
.05 
.67 
.05 
.67 

9.490286 
.490733 
.491180 
.491627 
.492073 
.492519 
.492965 
.493410 
.493854 
.494299 

7.45 
7.45 
7.45 
7.43 
7.43 
7.43 
7.42 
7.40 
7.42 
7.40 

10.509714 
.509867 
.508880 

.508373 
.507927 
.507481 
.507035 
.506590 
.506146 
.505701 

49 
48 
47 
46 
45 
44 
43 
48 
41 
40 

21 
22 
23 
24 

9.474519 
.474923 
.475327 
.475730 

6.73 
6.73 
6.72 

6r»»> 

9.979776 
.979737 
.979697 
.979658 

.65 
.67 
.65 

9.494743 
.495186 
.495630 
.496073 

7.38 
7.40 
7.38 

10.505257  39 
.50481  l 
.504370  37  ' 
.50890S 

25 
26 

27 
28 
29 

.476133 
.476536 
.476938 
.477340 
.477741 

.  i£ 

6.72 
6.70 
6.70 
6.68 
6  68 

.979618 
.979579 
.979539 
.979499 
.979459 

.67 
.65 
.67 
.67 
.67 

6K 

.496515 
.496957 
.497399 
.497841 
.498282 

7.37 
7.37 
7.37 
7.37 
7.35 

7  '45* 

.506486 

.50301 
.502601   :« 
.502159  32 
.501718  31 

30 

.478142 

6^67 

.979420 

0 

.67 

.498722 

i  .00 

7.35 

.501278 

30 

31 

9.478542 

6A7 

9.979380 

R7 

9.499163 

ft  qq 

10.500837  29 

32 

.478942 

.O< 

6tV7 

.979340 

.Of 

.499603 

I  .33 

.500397  28 

33 

.479342 

.  Oi 

.979300 

.67 

.500042 

i  .91 

.499958  27 

34 

.479741 

6.65 

6  AX 

.979260 

.67 

.500481 

7.32 

f  qo 

.499519  26 

35 

36 
37 

.480140 
.480539 
.480937 

.DO 

6.65 
6.63 

.979220 
.979180 
.979140 

167 
.67 

.500920 
.501359 
.501797 

1  •'. 

7.32 
7.30 

.499080  25 
.498641   24 
.498203  23 

38 

.481334 

6.62 

61-.) 

.979100 

.67 

no 

.502235 

7.30 

7  9ft 

.497765  22 

39 

40 

.481731 
.482128 

.0/0 
6.62 
6.62 

.979059 
.979019 

.Do 

.67 
.67 

.502672 
.503109 

<  .*o 

7.28 
7.28 

.49732S  -l 
.496891   -JO 

41 

9.482525 

6  Art 

9.978979 

fty 

9.503546 

7  27 

10.496454  1!) 

42 
43 

.482921 
.483316 

.OU 

6.58 

.978939 
.978898 

.O« 

.68 

.503982 
.504418 

7^27 

.496018  18 
.49558-3   17 

41 

.483712 

6.60 

.978858 

.67 

AP. 

.504854 

7.27 

r»  OK 

.495146  16 

45 
46 
47 

48 

.484107 
.484501 
.484895 
.485289 

6.58 
6.57 
6.57 
6.57 

.978817 
.978777 
.978737 
.978696 

.DO 

.67 
.67 
.68 

.505289 
.505724 
.506159 
.506593 

i  .  -•> 

7.25 
7.25 

7.23 

7.).) 

.494711   15 

.4!ir.'Ti;  it 
.493841  13 
.493407  12 

49 

.485682 

6.55 

.978655 

.68 

.507027 

.*B 

;  11 

50 

.486075 

6.55 
6.53 

.978615 

.67 
.68 

.507460 

7.'22 

;•)  10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.486467 
.486860 
.487251 
.487643 
.488034 
.488424 
.488814 
.489204 
.489593 
9.489982 

6.55 
6.52 
6.53 
6.52 
6.50 
6.50 
6.50 
6.48 
6.48 

9.978574 
.978533 
.978493 
.978452 
.978411 
.978370 
978329 
.978288 
.978247 
9.978206 

.68 
.67 
.68 
.68 
.68 
.68 
.68 
.68 
.68 

9.507893 
.508326 
.508759 
.509191 
.509622 
.510054 
.510485 
.510916 
.511346 
9.511776 

7.22 
7.23 
7.20 
7.18 
7.20 
7.18- 
7.18 
7.17 
7.17 

10.492107 
.491674 
.  J'.M-JH 
.490809 

.  I1MKJ7S 

.489946 
.489515 
.489084 
.488654 
10.4688M 

9 
8 
7 
6 
5 
4 
3 
2 

0 

' 

Cosine. 

D  r. 

Sine. 

D.  1". 

Cotang. 

D.  1'. 

Tang. 

' 

107- 


193 


72- 


18° 


TABLE  XII. — LOGARITHMIC    SINES, 


161° 


' 

Sine. 

D.  1". 

Cosine. 

D.I". 

Tang. 

D.  1". 

Cotang. 

' 

0  9.489982 

9.978206 

pQ 

9.511776 

r»  -j  7 

10.488224 

60 

1   .490371 

6.48 

.978165 

.Do 
fift 

.512206 

I  .  1  1 

.487794 

59 

2  j  .490759 

6.47 

.978124 

.Do 
Aft 

.512635 

71^ 

.487365 

58 

3   .491147 

6.47 

.978083 

.DO 

.513064 

t  .  1O 
71  K. 

.486936 

57 

4   .491535 

6.47 

.978042 

fift 

.513493 

.  lo 

71°. 

.486507 

56 

5   .491922 

6.45 

.978001 

.Do 
r-A 

.513921 

.  lo 

7  -jq 

.486079 

55 

6 

7 
8 
9 

.492308 
.492695 
.493081 
.493466 

6.43 
6.45 
6.43 
6.42 

.977959 
.977918 

.977877 
.977835 

.  t\J 

.68 
.68 
.70 

.514349 
.514777 
.515201 
.515631 

«  .  lo 

7.13 

7.12 
7.12 

.485651 
.485223 
.484796 
.484369 

54 
53 
52 
51 

10 

.493851 

6.42 
6.42 

.977794 

!70 

.516057 

7.12 

.483943 

50 

11 

9.494236 

649 

9.977752 

68 

9.516484 

r.  1rt  i  10.483516 

49 

12 

.494621 

.4/4 
A  4ft 

.977711 

7ft 

.516910 

,1'Ao     .483090 

48 

13 

.495005 

D.4U 

6qO 

.977669 

.  <U 

.517335 

£•  JQ     .482665 

47 

14 

.495388 

.OO 

.977628 

•JJJ- 

.517761 

r>  AQ 

.482239 

46 

15 
16 

.495772 
.496154 

6.40 
6.37 

6qo 

.977586 
.977544 

.  <0 
.70 

Aft 

.518186 
.518610 

<  .UO 

7.07 

7  07 

.481814 
.481390 

45 
44 

17 

.496537 

.00 
6°.7 

.977503 

.Do 

.519034 

r-  ryy 

.480966 

43 

18 

.496919 

.of 

.977461 

'r-A 

.519458 

t  .U< 

.480542 

42 

19 

.497301 

6.37 

6qK 

.977419 

.  <o 

I~A 

.519882 

7  ft^. 

.480118 

41 

20 

.497682 

.oO 

6.35 

.977377 

.  <u 
.70 

.520305 

t  .UO 

7.05 

.479695  40 

21 

9.498064 

6qq 

9.977335 

i  9.520728 

10.479272 

39 

22 

.498444 

.OO 

6qK 

.977293 

70 

.521151 

7  03 

.478849 

38 

23 

.498825 

.OO 

.977251 

.521573 

.478427 

37 

24 

.499204 

6.32 

6qq 

.977209 

'7ft 

.521995 

7.03 
7  03 

.478005 

36 

25 

.499584 

.00 
A  °.9 

.977167 

.  «u 
fffk 

.522417 

7  ft9 

.477583 

35 

26 

.499963 

u.o/* 

.977125 

.  i(J 

.522838 

<  .U^ 

.477162 

34 

27 

.500342 

6.32 

6qo 

.977083 

.70 

7ft 

.523259 

7  02 

.476741 

33 

28 

.500721 

.Ow 

6qft 

.977041 

.  <U 

.523680 

7  ftft 

.476320 

32 

29 

.501099 

.oil 

690 

.976999 

'•~n 

.524100 

i  .UU 
7  ftft 

.475900 

31 

30 

.501476 

,<£O 

6.30 

.976957 

!72 

.524520 

i  .UU 

7.00 

.475480 

30 

31 

9.501854 

69ft 

9.976914 

9.524940 

no 

10.475060 

29 

32 

.502231 

.do 

6  27 

.976872 

7ft 

.525359 

6  98 

.474641 

28 

33 

.502607 

69ft 

.976830 

.  i  U 

.525778 

6  no 

.474222 

27 

34 

.502984 

.do 
697 

.976787 

'r-A 

.526197 

.00 

.473803 

26 

35 
36 
37 
38 
39 

.503360 
.503735 
.504110 
.504485 
.504860 

.iff 

6.25 
6.25 
6.25 
6.25 

.976745 
.976702 
.976660 
.976617 
.976574 

.  <o 

.72 
.70 

.72 
.72 

.526615 
.527033 
.527451 

.527868 
.528285 

6^97 
6.97 
6.95 
6.95 

.473385 
.472967 
.472549 
.472132 
.471715 

25 
24 
23 
22 
21 

40 

.505234 

6^23 

.976532 

.70 

.72 

.528702 

6.95 
6.95 

.471298 

20 

41 

9.505608 

OO 

9.976489 

r-O 

9.529119 

10.470881 

19 

42 
43 
44 
45 
46 

.505981 
.506354 
.506727 
.507099 
.507471 

e!22 

6.22 
6.20 
6.20 

.976446 
.976404 
.976361 
.976318 
.976275 

'.70 

.72 
.72 
.72 

.529535 
.529951 
.530366 
.530781 
.531196 

b.9o 
6.93 
6.92 
6.92 
6.92 

.470465 
.470049 
.469634 
.469219 
.468804 

18 
17 
16 
15 
14 

47 
48 
49 

.507843 
.508214 
.508585 

6^18 
6.18 
61ft 

.976232 
.976189 
.976146 

.72 
.72 
.72 

.531611 
.532025 
.532439 

6.92 
6.90 
6.90 

.468389 
.467975 
.467561 

13 

12 
11 

50 

.508956 

.  lo 

6.17 

.976103 

.72 
.72 

.532853 

6.90 
6.88 

.467147 

10 

51 

9.509326 

617 

9.976060 

9.533266 

6QQ 

10.466734 

9 

52 

.509696 

.  1  • 

6  15 

.976017 

.72 

79 

.533679 

.OO 
600 

.466321 

8 

53 

.510065 

.975974 

.  i  A 

79. 

.534092 

.  GO 

.465908 

7 

54 
55 
56 

.510434 
.510803 
.511172 

e!is 

6.15 
6-«q 

.975930 

.975887 
.97'5844 

.  (6 

.72 
.72 

.534504 
.534916 
.535328 

6.87 
6.87 
6.87 

.465496 
.465084 
.464672 

6 
5 
4 

57 

.511540 

.  lo 
6  12 

.975800 

.73 

.535739 

6.85 

6QK 

.464261 

3 

58 

.511907 

61  °. 

.975757 

"An 

.536150 

.00 

.463850 

2 

59 
60 

.512275 
9.512642 

.  lo 

6.12 

.975714    -Ao 
9.975670  j   -'d 

.536561 
9.536972 

6.85 
6.85 

.463439 
10.463028 

1 

0 

'  1  Cosine. 

D.  1'. 

Sine.    D.  1*. 

Cotang. 

D.  r. 

Tang. 

' 

108* 


194 


71° 


COSINES,    TANGENTS,    AND    COTANGENTS.  160° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

9.512642 

6  12 

9.975670 

72 

9.536972 

6QQ 

10.463028 

60 

1 

.513009 

.975627 

r-o 

.537382 

.00 

.462618 

59 

2 

.513375 

A  ift 

.975583 

.  <•> 
r  o 

.537792 

6.83 

.462208 

58 

3 

.513741 

6  10 

.975539 

.  1  O 

.538202 

6.83 

.461798 

57 

4 

.514107 

.975496 

'r-o 

.538611 

6.82 

.461389 

56 

5 

6 

.514472 
.514837 

6  .  08 
6.08 

6flK 

.975452 
.975408 

.73 
.73 

.539020 
.539429 

6.82 
6.82 

.460980 
.460571 

55 

54 

7 
8 
9 

.515202 
.515566 
.515930 

.Uo 

6.07 
6.07 

6/17 

.975365 
.975321 
.975277 

.73 
.73 

r-o 

.539837 
.540245 
.540653 

6.80 
6.80 
6.80 

.460163 
.459755 
.459347 

53 
52 
51 

10 

.516294 

.Ui 

6.05 

.975233 

.  to 

.73 

.541061 

6.80 
6.78 

.458939 

50 

11 

9.516657 
.517020 

6.05 

9.975189 
.975145 

.73 

9.541468 
.541875 

6.78 

10.458532 
.458125 

49 

48 

13 

.517382 

6.03 

.975101 

.73 

.542281 

6.77 

.457719 

47 

14 
15 

.517745 
.518107 

6.05 
6.03 

.975057 
.975013 

.73 
.73 

.542688 
.543094 

6.78 
6.77 

.457312 
.456906 

46 
45 

16 

.518468 

6.02 

.974969 

.73 

.543499 

6.75 

.456501 

44 

17 

.518829 

6.02 

.974925 

.73 

.543905 

6.77 

.456095 

43 

18 

.519190 

6.02 

.974880 

.75 

.544310 

6.75 

.455690 

42 

19 

.519551 

6.02 

.974836 

.73 

.544715 

6.75 

41 

20 

.519911 

6.00 

.974792 

.73 

.545119 

6.73 

'.454881 

40 

6.00 

.73 

6.75 

21 
22 

9.520271 
.520631 

6.00 

9.974748 
.974703 

.75 

9.545524 
.545928 

6.73 

10.454476 

.454072 

39 

38 

23 

.520990 

5.98 

.974659 

.73 

.546331 

6.72 

.453669 

37 

24 

.521349 

5.98 

.974614 

.75 

.546735 

6.73 

.453265 

36 

25 

.521707 

5.97 

.974570 

.73 

.547138 

6.72 

.452862 

35 

26 

27 
28 

.522066 
.522424 
.522781 

5.98 
5.97 
5.95 

.974525 
.974481 
.974436 

.75 
.73 
.75 

.547540 
.547943 
.548345 

6.70 

6.72 
6.70 

.452460 
.452057 
.451655 

34 
33 
32 

29 

.523138 

5.95 

.974391 

.75 

.548747 

6.70 

.451253 

31 

30 

.523495 

5.95 

.974347 

.73 

.549149 

6.70 

.450851 

30 

5.95 

.75 

6.68 

31 

9.523852 

9.974302 

9.549550 

10.450450 

29 

32 

.524208 

5.93 

.974257 

.75 

.549951 

6.68 

.450049 

28 

33 

34 

.524564 
.524920 

5.93 
5.93 

.974212 
.974167 

.75 
.75 

.550352 
.550752 

6.68 
6.67 

.449648 
.449248 

27 
26 

35 

36 

.525275 
.525630 

5.92 
5.92 

.974122 
.974077 

.75 
.75 

.551153 
.551552 

6.68 
6.65 

.448847 
.448448 

25 
24 

37 

38 

.525984 
.526339 

5.90 
5.92 

.974032 
.973987 

.75 

.75 

.551952 
.552351 

6.67 
6.65 

.448048 
.447649 

23 
22 

39 
40 

.526693 
.527046 

5.90 
5.88 
5.90 

.973942 
.973897 

.75 
.75 
.75 

.552750 
.553149 

6.65 
6.65 
6.65 

.447250 
.446851 

21 

20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.527400 
.527753 
.528105 
.528458 
.528810 
.529161 
.529513 
.529864 
.530215 
.530565 

5.88 
5.87 
5.88 
5.87 
5.85 
5.87 
5.85 
5.85 

5.  as 

5.83 

9.973852 
.973807 
.973761 
.973716 
1  .973671 
.973625 
.973580 
.973535 
.973489 
.973444 

.75 
.77 
.75 
.75 
.77 
.75 
.75 
.77 
.75 
.77 

9.553548 
.553946 
.554344 
.554741 
.555139 
.55f.536 
.555933 
.556329 
.556725 
.557121 

6.63 
6.63 
6.62 
6.63 
6.62 
6.62 
6.60 
6.60 
6.60 
6.60 

10.446452 
.446054 
.445656 
.445259 
.444861 
.444464 
.444067 
.443671 
.443275 
.442879 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.530915 
.531265 
.531614 
.531963 
.532312 
.532661 
.533009 
.533357 
.533704 
9.534052 

5.83 

5.82 
5.82 
5.82 
5.82 
5.80 
5.80 
5.78 
5.80 

9.973398 
.973352 
.973307 
.973261 
.973215 
.973169 
.973124 
.973078 
.973032 
9.972986 

.77 
.75 

.77 
.77 
.77 
.75 
.77 
.77 
.77 

9.557517 
.557913 
.558308 
.558703 
.559097 
.559491 
.559885 
.560279 
.560673 
9.561066 

6.60 
6.58 
6.58 
6.57 
6.57 
6.57 
6.57 
6.57 
6.55 

10.442483 
.442087 
.441692 
.441297 
.440903 
.440509 
.440115 
.439721 
.439327 
10.488984 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  1'. 

Tang,  i  ' 

109' 


195 


70' 


20° 


TABLE   XII. — LOGARITHMIC    SINES, 


159° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1*. 

Tang. 

D.  r. 

Cotang. 

• 

0 

1 

9.534052 
534399 

5.78 

9.972986 
.972940 

.77 

9  561066 
.561459 

6  55 

6  tq 

10.438934 
.438541 

60 
59 

2 
3 

.534745 
.535092 

5  77 

5.78 

.972894 
.972848 

'77 

77 

561851 
562244 

Do 

6.55 

6  CO 

438149 
.437756 

58 
57 

4 

.535438 

5.77 

972802 

.  I  ( 

78 

.562636 

.Do 

6  53 

.437364 

56 

5 

.535783 

5.75 

972755 

<o 

.563028 

.436972 

55 

6 

.536129 

5.77 

.972709 

77 

563419 

6  tq 

.436581 

54 

7 

.536474 

5  75 

.972663 

.  i  1 

563811 

Do 

436180 

53 

8 

.536818 

5.73 

972617 

78 

564202 

6  52 

o  to 

.435798 

52 

9 
10 

.537163 
.537507 

5!  73 
5.73 

9r<  2570 
.972524 

.  <o 

,77 

564593 
564983 

D.  D/* 

6  50 
6.50 

,435407 
.435017 

51 
50 

11 
12 
13 
14 

9.537851 
538194 
538538 
536880 

5.72 
5  73 
5.70 

9.972478 
972431 
972385 
972338 

.78 

.77 
.78 

r-o 

9-565373 
565763 
.566153 
566542 

6.50 
6  50 
6.48 

6KA 

10.434627 
.434237 
.433847 
,433458 

49 

48 
47 
46 

15 

.539223 

5.72 

.972291 

.  1  O 

77 

.566932 

.OU 

6  47 

.433068 

45 

16 

539565 

5.70 

.972245 

<  t 
78 

.567320 

648 

.432680 

44 

17 
18 

.539907 
.540249 

5.70 
5.68 

972198 
.9721M 

.  <o 

.78 

567709 

.568098 

.<*O 

6.48 

647 

.432291 
.431902 

43 

42 

19 
20 

.540590 
.540931 

5.68 
5.68 
5.68 

.972105 
.972058 

i78 

568486 
568873 

.4< 

6.45 
6.47 

.431514 
.431127 

41 
40 

21 

9.541272 

9.972011 

r-o 

-9.569261 

10.430739 

39 

22 

23 

.541613 
.541953 

5.68 
5.67 

.971964 
971917 

.  <o 

.78 

.569648 
.570035 

6^45 

.430352 
.429965 

38 
37 

24 

.542293 

5.67 

.971870 

.78 

.570422 

6.45 
6  45 

,429578 

36 

25 

.542632 

5.65 

.971823 

r-o 

.570809 

64P. 

.429191 

35 

~6 

.542971 

5.65 

.971776 

.  iO 
78 

.571195 

.'lo 
64'} 

.428805 

34 

27 
28 
29 
30 

.543310 
543649 
.543987 
.544325 

5  '.65 
5.63 
5.63 
5.63 

.971729 
.971682 
.971635 
.971588 

.  <o 

.78 
.78 
.78 
.80 

.571581 
.571967 
.572352 
572738 

•*o 
6.4S 
6.42 
6.43 
6.42 

,428419 
.428033 
.427648 
.427262 

33 
32 
31 
30 

31 
32 
33 
34 
35 

9.544663 
.545000 
.545338 
.545674 
.546011 

5.62 
5.63 
5.60 
5.62 

9.971540 
.971493 
.971446 
.971398 
.971351 

.78 
.78 
.80 
.78 

9.573123 

.573507 
.573892 
.574276 
.574660 

6  40 
6.42 
6.40 
6.40 

10.426877 
.426493 
.426108 
.425724 
.425340 

29 

28 
27 
26 
25 

36 
37 
38 

546347 
.546683 
.547019 

5.60 
5.60 
5.60 

.971303 
.971256 

.971208 

!78 
.80 

.575044 

.575427 
.575810 

6^38 
6  38 

6  op 

.424956 
.424573 
.424190 

24 
23 

22 

39 
40 

.547354 
.547669 

5.58 
5.58 
5.58 

.971161 
.971113 

.78 
80 
.78 

.576193 
.576576 

.00 

6.38 
6.38 

.423807 
.423424 

21 
20 

41 

9.548024 

Sfcrp 

9.971066 

PA 

9.576959 

6  37 

10.423041 

19 

42 

.  548359 

.OO 

.971018 

.oU 
PA 

.577341 

6  37 

.422659 

18 

43 

.548693 

5  57 

.970970 

.oU 

OA 

.577723 

.422277   17 

44 

.549027 

5.57 

.970922 

.oU 

.578104 

607 

.421896 

16 

45 

46 

.549360 
.549693 

5.55 
5.55 

5fK 

.970874 
.970827 

.80 
.78 

PA 

.578486 
.578867 

.o< 
6.35 
6  35 

.421514 
.421133 

15 
14 

47 

48 

.550026 
.550359 

.OO 

5.55 

pr  te 

.970779 
.970731 

.  oU 

.80 

.579248 
.579629 

6^35 
6  33 

.420752 
.420371 

13 
12 

49 
50 

.550692 
.551024 

O  .DD 

5.53 
5.53 

.970683 
.970635 

.'80 
.82 

.580009 
580389 

eias 

6.33 

.419991 
.419611 

11 
10 

61 

9.551356 

9.970586 

OA 

9.580769 

6qq 

10.419231 

9 

52 

.551687 

5.52 

.970538 

.oU 

.581149 

.00 

.418851 

8 

53 
54 

.552018 
.552349 

5.52 
5.52 

.970490 
.970442 

.80 
.80 

,581528 
581907 

6.32 
6.32 

.418472 
.418093 

7 
6 

55 

.552680 

5.52 

.970394 

.80 

.582286 

6.32 

.417714 

5 

56 

.553010 

5.50 

.970345 

.82 

PA 

.582665 

6.32 
69.9 

.417335 

4 

57 
58 

.553341 
.553670 

5.  '48 

5tA 

.970297 
.970249 

.OU 

.80 

.583044 
.583422 

.  O/i 

6.30 
6  30 

.416956 
.416578 

3 
2 

59 

.554000 

.OU 

.970200 

•5? 

.583800 

.416200 

1 

60 

9.554329 

5.48 

9.970152 

.80 

9.584177 

6.28 

10.415823 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1".  1 

Cotang. 

D.  i". 

Tang. 

' 

110° 


196 


21° 


COSINES,    TANGENTS,    AND    COTANGENTS.  168° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1-. 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.554329 
.554658 
.554987 
.555315 
.555643 
.555971 
.556299 
.556626 
.556953 
.557280 
.557606 

5.48 
5.48 
5.47 
5.47 
5.47 
5.47 
5.45 
5.45 
5.45 
5.43 
5.43 

9.970152 

.970103 
.  97  0055 
.970006 
.909957 
.969909 
.909800 
.969811 
.969762 
.969714 
.969665 

.82 
.SO 
.82 
.82  j 
.80  i 
.82 
.82 
.82 
.80 
.82 
.82 

9.584177 
.584555 
.584982 
.585809 
.585686 
.580002 
.586439 
.586815 
.587190 
.587566 
.5S7941 

6.30 
6.28 
6.28 
6.28 
6.27 
6.28 
6.27 
6.25 
6.27 
6.25 
6.25 

10.415823 
.415446 

.414601 

.414314 
.413938 
.413561 
.413185 
.412810 
.412484 
.412069 

60 

66 

57 
56 
55 
54 
53 
52 
51 
60 

11 
12 
13 
14 
15 
16 

9.557932 
.558258 
.558583 
.558909 
.559234 
.559558 

5.43 
5.42 
5.43 
5.42 
5.40 
5.42 

9.969616 
.969567 
.969518 
.969469 
.969420 
.969370 

.82 
.82 
.82 
.82 
.83 

9.588316 
.588691 
.589066 
.589440 
.589814 
.590188 

6.25 
6.25 
6.23 
6.23 
6.23 

10.411684 
.411809 
.410984 
.410660 

.410186 

49 

48 
47 
40 
45 
44 

17 

18 
19 
20 

.559883 
.560207 
.560531 
.560855 

5.  40 
5.40 
5.40 
5.38 

.969321 
.969272 
.969223 
.969173 

!82 
.82 

.as 

.82 

.590562 
.590935 
.591308 
.591681 

6^22 
6.22 
6.22 
6.22 

.409438 
.409005 
.408692 
.408319 

43 
42 

41 
40 

21 
22 
23 
24 

9.561178 
.561501 
.561824 
.562146 

5.38 
5.38 
5.37 
5  37 

9.969124 
.969075 
.969025 
.968976 

.82 
.83 
.82 

oq 

9.592054 
.592426 
.592799 
.593171 

6.20 
6.22 
6.20 

10.407946 
.407574 
.407201 
.406829 

39 
88 
37 
36 

25 
26 

27 

.562468 
.562790 
.563112 

5^37  ' 
5.37 

5  OK 

.968926 
.968877 
.968827 

.OO 

.82 
.83 

.593542 
.593914 
.594285 

6.18 
6.20 
6.18 

.406458 
.40608(5 
.405715 

35 
84 

M 

28 

.563433 

.OO 

5q7 

.968777 

.83 

.594656 

6.18 

.405344 

29 

.563755 

.O< 

.968728 

.82 

.595027 

6.18 

.404973 

31 

30 

.564075 

5.33 
5.35 

.968678 

.83 
.83 

.595398 

6.18 
6.17 

.404602 

30 

31 

9.564396 

K  qq 

9.968628 

9.595768 

10.404532 

29 

32 

.564716 

O.  oo 

.968578 

.83 

.596138 

6.17 

.403802  28 

33 

.565036 

5.33 

5qq 

.968528 

.83 

.596508 

6.17 

617 

.403492  57 

34 
35 
36 

.565356 
.565676 
.565995 

.00 

5.33 
5.32 

.968479 
.968429 
.968379 

.82 
.83 
.83 

.596878 
.597247 
.597616 

.It 

6.15 
6.15 

.403122  5ft 
.402753  2.1 
.4023K1   51 

37 

.566314 

5.32 

.968329 

.83 

.597985 

6.15 

.4020  K 

38 

.566632 

5.30 

.968278 

.85 

.598354 

6.15 

.40101' 

39 

.566951 

5.32 

.968228 

.83 

.598722 

6.13 

.401278 

21 

40 

.567269 

5.30 
5.30 

.968178 

.83 
.83 

.599C91 

6.15 
6.13 

.400909 

20 

41 
42 

43 
44 
45 
46 

9.567587 
.567904 
.568222 
.568539 
.568856 
.569172 

5.28 
5.30 
5.28 
5.28 
5.27 

597 

9.968128 
.968078 
.968027 
.967977 
.967927 
.967876 

.83 

.85 
.83 
.83  : 

.85  i 

QQ 

9.599459 
.599827 
.600194 
.600562 
.600929 
.601296 

6.13 
6.12 
6.13 
6.12 
6.12 
6  12 

10.400541 
.400173 
.399806 
.399438 
.399071 
.398704 

19 

1H 
17 
10 
15 
14 

47 
48 
49 

.569488 
.569804 
.570120 

.lilt 

5.27 
5.27 

.967826 
.967775 
.967725 

.OO 

.85 

.83 

.601663 
.602029 
.602395 

6.'  10 
6.10 
6  10 

.896897 

.397971 
.397605 

13 
12 
11 

50 

.570435 

5^27 

.967674 

!83 

.602761 

e!io 

.397239 

10 

61 

52 
53 
54 
55 
56 
57 

9.570751 
.571066 
.571380 
.571695 
.572009 
.572323 
.572636 

5.25 
5.23 
5.25 
5.23 
5.23 
5.22 

9.967624 
.967573 
.967522 
.967471 
.967421 
.967370 
.967319 

.85 
.85 
.85 
.83 
.85 
.85 

9.603127 
.603493 
.603858 
.604223 
.604588 
.604953 
.605317 

6.10 
6.08 
6.08 
6.08 
6.08 
6.07 
6  08 

10.396873 
.896607 

.396142 
.895777 
.896412 

.894688 

9 
8 
7 
6 
5 
4 
3 

58 

.572950 

5.23 

.967268 

•§3 

.605682 

2 

59 
60 

.573263 
9.573575 

5.22 
5.20 

.967217 
9.907100 

.85 

.85  ! 

.606046 
9.606410 

6.07 

10.393590 

1 
0 

' 

Cosine. 

D.  1". 

Sine.   D.  r.   Cotang.  D.  1'. 

Tang. 

L 

111° 


197 


68* 


22° 


TABLE   XII. — LOGARITHMIC   SINES, 


157° 


' 

Sine. 

D.  i*. 

Cosine. 

D.  1'. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

4 

9.573575 

.573888 
.574200 
.574512 

.574824 

5.22 
5.20 
5.20 
5.20 

59O 

9.967166 
.967115 
.967064 
.967013 
.966961 

.85 

.85 
.85 

.87 

QC 

9.606410 
.606773 
.607137 
.607500 
.607863 

6.05 
6.07 
6.05 
6.05 

A  no. 

10.393590 
.393227 
.392863 
.392500 
.392137 

60 

59 
58 
57 
56 

5 

.575136 

.<*U 

.966910 

.OO 

.608225 

O.Uo 

.391775 

55 

6 

8 
9 

.575447 
.575758 
.576069 
.576379 

5.18 
5.18 
5.18 
5.17 

517 

.966859 
.966808 
.966756 
.966705 

.85 
.85 
.87 
.85 

87 

.608588 
.608950 
.609312 
.609674 

6.05 
6.03 
6.03 
6.03 

6  no 

.391412 
.391050 
.390688 
.390326 

54 
53 
52 
51 

10 

.576689 

.  1  1 

5.17 

.966653 

.of 

.85 

.610036 

.Uo 

6.02 

.389964 

50 

11 

9.576999 

517 

9.966602 

87 

9.610397 

6AO 

10  389603 

49 

12 
13 
14 
15 

.577309 
.577618 

577927 
.578236 

.  1  1 

5.15 
5.15 
5  15 

.966550 
.966499 
.966447 
.966395 

•  of 

.85 
.87 
.87 

.610759 
.611120 
.611480 
.611841 

.Uo 

6.02 
6.00 
6.02 

.389241 
.388880 
.388520 
.388159 

48 
47 
46 
45 

16 
17 

.578545 

.578853 

5.15 
5.13 

5   -IK 

.966344 
.966292 

.85 
.87 

87 

.612201 
.  612561 

6.00 
6.00 

6AA 

.387799 
.387439 

44 
43 

18 

.579162 

.  10 

.966240 

.01 

.612921 

.UU 

.387079 

42 

19 
20 

.579470 

.579777 

5.13 
5.12 
5.13 

.966188 
.966136 

.87 
.87 
.85 

.613281 
.613641 

6.(;0 
6.00 
5.98 

.386719 
.386359 

41 
40 

21 

9.580085 

9.966085 

87 

9.614000 

10.386000 

39 

22 
23 

.580392 
.580699 

5  '.12 
5in 

.966033 
.965981 

.OY 

.87 

87 

.614359 
.614718 

5.'98 

5QQ 

.385641 

.385282 

38 
37 

24 

.581005 

.  1U 

.965929 

.01 

88 

.615077 

yo 

,384923 

36 

25 
26 
27 

28 
29 

.581312 
581618 
.581924 
.582229 
.582535 

5.12 
5.10 
5.10 
5.08 
5.10 

5AQ 

.965876 
.965824 
.965772 
.965720 
.965668 

.00 

.87 
.87 
.87 
.87 

88 

.615435 
.615793 
.616151 
.616509 
.616867 

5^97 
5.97 
5.97 
5.97 

K  QK 

.384565 
.384207 
.383849 
.383491 
.383133 

85 
34 
33 
32 
31 

30 

.582840 

.Uo 

5.08 

.965615 

.00 

.87 

.617224 

o  .  yo 
5  97 

.382776 

30 

31 
32 

9.583145 
.583449 

5.07 
5na 

9  965563 
.965511 

.87 

88 

9.617582 
617939 

5.95 

fr  QO. 

10  382418 
.382061 

29 

28 

33 
34 
35 
36 

583754 
.584058 
.584361 
.584665 

.Uo 
5.07 
5.05 
5.07 

5AK 

.  965458 
.965406 
965353 
.965301 

.OO 

.87 
.88 
.87 

QQ 

.618295 
.618652 
.619008 
.619364 

o.  yo 
5.95 
5.93 
5.93 

5QO 

.381705 
.381348 
.380992 
.380636 

27 
26 
25 
24 

37 

.584968 

.UO 
5O7 

.965248 

.OO 
QQ 

.619720 

.yo 

5  QO 

.380280 

23 

38 
39 

.585272 
.585574 

.U< 

5.03 

5AK 

.965195 
.965143 

.OO 

.87 

QQ 

.620076 
.620432 

.yo 
5  93 
5  92 

.379924 
.379568 

22 
21 

40 

.585877 

.UO 

5.03 

.965090 

.OO 

.88 

.620787 

5192 

.379213 

20 

41 

9.586179 

5A.K 

9.965037 

88 

9.621142 

10.378858 

19 

42 

.586482 

.UO 

.964984 

.00 
88 

.621497 

5  no 

.378503 

18 

43 
44 

586783 
.587085 

5.03 

.964931 
.964879 

.OO 

.87 

88 

.621852 
.622207 

.»/* 
5.92 
5  on 

.378148 
.377793 

17 
16 

45 

.587386 

5.02 

5AO 

.964826 

.OO 
88 

.622561 

.yu 

5  Of) 

.377439 

15 

46 

.587688 

.Uo 

.964773 

.00 

.622915 

.yu 

.377085 

14 

47 
48 
49 

.587989 
.588289 
.588590 

5.02 
5.00 
5.02 
5  00 

.964720 
.964668 
.964613 

.88 
.90 

.88 

QQ 

.623269 
.623623 
.623976 

5.90 
5.90 

5.88 

5QA 

.376731 
.376377 
.37'6024 

13 
12 
11 

50 

.588890 

5!oo 

.964560 

.OO 

.88 

.624330 

.yu 
5.88 

.375670 

10 

51 

9.589190 

4  no 

9.964507 

QQ 

9.624683 

5  CO 

10.375317 

9 

52 
53 

.589489 
.589789 

.yo 
5.00 

4Q8 

.964454 
.964400 

.OO 

.90 

.625036 
.625388 

.00 
5.87 

.374964 
.374612 

8 

7 

54 

.590088 

.yo 

.964347 

.88 

.625741 

5.88 

.374259 

6 

55 

56 

.590387 
.590686 

4.98 
4.98 

.964294 
.964240 

.88 
.90 

.626093 
.626445 

5.87 
5.87 

.373907 
.373555 

5 

4 

57 
58 

.590984 
591282 

4.97 
4.97 

4Q7 

.964187 
.964133 

.88 
.90 

88 

.626797 
.627149 

5.87 
5.87 

587 

.373203 
.372851 

3 

2 

59 
60 

.591580 
9.591878 

.y« 
4.97 

.964080 
9.964026 

.00 

.90 

.627501 

9.627852 

.of 

5.85 

.372499 
10.372148 

1 
0 

' 

Cosine. 

D.  1". 

I  Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang. 

~ 

198 


67° 


23° 


COSINES,    TANGENTS,    AND    COTANGENTS.  166° 


' 

Sine. 

D.  1*. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 

3 

4 
5 
6 
7 
8 
9 
10 

9.591878 
.592176 
.592473 
.592770 
.593067 
.593363 
.593659 
.593955 
.594251 
.594547 
.594842 

4.97 
4.95 
4.95 
4.95 
4.93 
4.93 
4.93 
4.93 
4.93 
4.92 
4.92 

9.964026 
.963972 
.963919 
.963865 
.963811 
.963757 
.963704 
.963650 
.963596 
.963542 
.963488 

.00 
.88 
.90 
.90 
.90 
.88 
.90 
.90 
.90 
.90 
.90 

9.627863 
.688808 
.628554 
.628905 
.629856 
.629606 
.629956 
.630306 
.630656 
.631005 
.631355 

5.85 
5.85 
5.85 
5.83 
5.85 
5.83 
5.83 
5.83 
5.89 
5.83 
5.82 

10.  372  118 
,871707 
.871440 
.871095 
.870745 
.370394 
.370044 
.869604 
.369344 
.868995 
.368645 

80 

59 

58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.595137 
.595432 
.595727 
.596021 
.596315 
.596609 
.596903 
.597196 
.597490 
.597783 

4.92 
4.92 
4.90 
4.90 
4.90 
4.90 
4.88 
4.90 
4.88 
4.87 

9.963434 
.963379 
.963325 
.963271 
.963217 
.963163 
.963108 
.963054 
.962999 
.962945 

.92 
.90 
.90 
.90 
.90 
.92 
.90 
.92 
.90 
.92 

9.631704 
.632053 
.632402 
!  .632750 
.633099 
.633447 
.633795 
.634143 
.634490 
.634838 

5.82 
5.82 
5.80 
5.82 
5.  SO 
5.80 
5.80 
5.78 
5.80 
5.78 

10.368296 
.887947 
.867598 

.367250 
.366901 
.366553 
.366205 
.865857 
.365510 
.365162 

49 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

23 
24 
25 

9.598075 
.598368 
.598660 
.598952 
.599244 

4.88 
4.87 
4.87 

4.87 
407 

9.962890 
.962836 
.962781 
.962727 
.962672 

.90 
.92 
.90 
.92 

9.635185 
.635532 
.635879 
.636226 
.636572 

5.78 
5.78 
5.78 
5.77 

10.364815 
.864468 

.364121 
.363774 
.363428 

39 
38 
37 
36 
35 

26 
27 

28 
29 

.599536 
.599827 
.600118 
.600409 

:.OI 

4.85 
4.85 
4.85 
4  85 

.962617 
.962562 
.962508  . 
.962453 

.92 
.92 
.90 
.92 

.636919 
.637265 
.637611 
.637956 

5.78 
5.77 
5.77 
5.75 

5r>ry 

.363081 
.362735 
.362389 
.362044 

34 
33 

32 
31 

30 

.600700 

4.83 

.962398 

!92 

.638302 

.  It 

5.75 

.361698 

30 

31 

9.600990 

4  83 

9.962343 

9.638647 

57K 

10.361353 

29 

32 
33 

.601280 
.601570 

4^83 

400 

.962288 
.962233 

!92 

.638992 
.639337 

.  1  •) 

5.75 

.361008- 
.360663 

28 
27 

34 

.601860 

.OO 
4QQ 

.962178 

.92 

.639682 

5.75 

5<~K 

.360318 

26 

35 

36 
37 

38 

.602150 
.602439 
.602728 
.603017 

.oO 

4.82 
4.82 
4.82 

480 

.962123 
.962067 
.962012 
.961957 

!93 
.92 
.92 

.640027 
.640371 
.640716 
.641060 

.  1.) 

5.73 
5.75 
5.73 

5r»O 

.359973 
.359629 
.359284 
.358940 

25 
24 
23 
22 

39 

.603305 

.  of 

.961902 

•~| 

.641404 

.  1  •  ) 

.358596 

21 

40 

.603594 

4^80 

.961846 

!92 

.641747 

5173 

.358253 

20 

41 

9.603882 

4  on 

9.961791 

9.642091 

10.357909 

19 

42 

.604170 

.oU 

.961735 

•*2 

.642434 

5.72 

.357566 

18 

43 

.604457 

4.78 

.961680 

.92 

.642777 

5.72 

5r»O 

.35?223 

17 

44 
45 
46 
47 
48 

.604745 
.605032 
.605319 
.605606 
.605892 

4.78 
4.78 
4.78 
4.77 

.961624 
.961569 
.961513 
.961458 
.961402 

!92 
.93 
.92 
.93 

.643120 
.643463 
.643806 
.644148 
.644490 

.  i  ~ 

5.72 
5.72 
5.70 
5.70 

.356880 
.356537 

!  855862 
.355510 

16 
15 
14 
13 
12 

49 
50 

.606179 
.606465 

4.78 
4.77 

4.77 

.961346 
.961290 

.93 
.93 
.92 

.644832 
.645174 

5.70 
5.70 
5.70 

.355168 

.354S-G 

11 
10 

51 

9.606751 

9.961235 

9.645516 

5(*Q 

10.354484 

9 

52 

.607036 

4.75 

.961179 

.93 

.645857 

.OO 

.854149 

8 

53 
54 
55 
56 

.607322 
.607607 
.607892 
.608177 

4.77 
4.75 
4.75 
4.75 

.961123 
.961067 
.961011 
.960955 

.93 
.93 
.93 
.93 

.646199 
.646540 
.646881 
.647222 

5.70 
5.68 
5.68 
5.68 

5K7 

.858801 
.858440 

.353119 
.858778 

7 
6 
5 
4 

57 

.608461 

4.73 

.960899 

.93 

.647562 

.Ol 
5  Aft 

3 

58 
59 
60 

.608745 
.609029 
9.609313 

4.73 
4.73 
4.73 

.960843 
.960786 
9.960730 

.93 
.95 
.93 

.647903 
.648243 
9.648583 

.08 

5.67 
5.67 

.858097 

.351757 
10.351417 

2 

1 
0 

9 

Cosine. 

D.  1'. 

Sine. 

D.  1".  |i  Cotang. 

D.  r. 

Tang. 

' 

113' 


199 


66- 


24° 


TABLE   XII. LOGARITHMIC    SINES, 


155° 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

i 

0 

1 

9.609313 
.609597 

4.73 

9.960730 
.960674 

.93 

9.648583 

.648923 

5.67 

10.351417 
.351077 

60 
59 

2 

.609880 

4.  72 

4*~q 

.960618 

.93 
05 

.649263 

5.67 

.350737  !  58 

3 

.610164 

.  IO 

.960561 

.649602 

5  ,  65 

.350398  57 

4 
5 

.610447 
.610729 

4.72 
4.70 

.960505 
.960448 

!95 

.649942 
.650281 

5.67 
5.65 

.350058 
.349719 

56 
55 

6 

.611012 

4.72 

47rt 

.960392 

.93 

QK. 

.650620 

5.65 

.349380 

54 

7 

.611294 

.  t(J 

.960335 

.yo 

.650959 

5.65 

.349041 

53 

8 
9 
10 

.611576 
.611858 
.612140 

4.70 
4.70 
4.68 

.960279 
.960222 
.960165 

.93 
.95 
.95 
.93 

.651297 
.651636 
.651974 

5^65 
5.63 
5.63 

.348703 
.348364 
.348026 

52 
51 
50 

11 

9.612421 

4  CQ 

9.960109 

QK 

9.652312 

5f>O 

10.347688 

49 

12 
13 
14 

.612702 
.612983 
.613264 

.Do 

4.68 
4.68 

4  CO 

.960052 
.959995 
.959938 

.yo 
.95 
.95 

QO 

.652650 
.652988 
.65a326 

.Do 

5.63 
5.63 

.347350 
.347012 
.346674 

48 
47 
46 

15 
16 
17 

.613545 
.613825 
.614105 

.  Do 

4.67 
4.67 

4K7 

.959882 
.959825 
.959768 

.yo 
.95 
.95 

QK 

.653663 
.654000 
.654337 

5^62 
5.62 

.346337 
.346000 
.345663 

45 
44 
43 

18 
19 
20 

.614385 
.614665 
.614944 

.Ol 

4.67 
4.65 
4.65 

.959711 
.959654 
.959596 

.yo 

.95 
.97 
.95 

.654674 
.  .655011 
.655348 

5.62 
5.62 
5.62 
5.60 

.345326 
.344989 
.344652 

42 
41 
40 

21 

22 
23 
24 
25 
26 
27 
28 
29 
30 

9.615223 
.615502 
.615781 
.616060 
.616338 
.616616 
.616894 
.617172 
.617450 
.617727 

4.65 
4.65 
4.65 
4.63 
4.63 
4.63 
4.63 
4.63 
4.62 
4.62 

9.959539 
.959482 
.959425 
.959368 
.959310 
.959253 
.959195 
.959138 
.959080 
.959023 

.95 
.95 
.95 
.97 
.95 
.97 
.95 
.97 
.95 
.97 

9.655684 
.656020 
.656358 
.656692 
.657028 
.657364 
.657699 
.658034 
.658369 
.658704 

5.60 
5.60 
5.60 
5.60 
5.60 
5.58 
5.58 
5.58 
5.58 
5.58 

10.344316 
.343980 
.343644 
.343308 
.342972 
.342636 
.342301 
.341966 
.341631 
.341296 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

9.618004 
.618S81 
.618558 
.618834 
.619110 
.619386 
.619662 
.619938 
.620213 

4.62 
4.62 
4.60 
4.60 
4.60 
4.60 
4.60 
4.58 

9.958965 
.958908 
.958850 
.958792 
.958734 
.958677 
.958619 
.958561 
.958503 

.95 
.97 
.97 
.97 
.95 
.97 
.97 
.97 

9.659039 
.659373 
.659708 
.660042 
.660376 
.660710 
.661043 
.661377 
.661710 

5.57 
5.58 
5.57 
5.57 
5.57 
5.55 
5.57 
5.55 

10.340961 
.340627 
.340292 
.339958 
.339624 
.339290 
.338957 
.338623 
.338290 

29 
28 
27 
26 
25 
24 
23 
22 
21 

40 

.620488 

4.58 
4.58 

.958445 

.97 
.97 

.662043 

5.55 
5.55 

.337957 

20 

41 
42 
43 
44 

9.620763 
.621038 
.621313 
.621587 

4.58 
4.58 
4.57 

9.958387 
.958329 
.958271 
.958213 

.97 
.97 
.97 

9.662376 
.662709 
.663042 
.663375 

5.55 
5.55 
5.55 

10.337624 
.337291 
.336958 
.336625 

19 

18 
17 
16 

45 

.621861 

4.57 

.958154 

*]5» 

.663707 

5.53 

.336293 

15 

46 
47 

.622135 
.622409 

4  '.87 

.958096 
.958038 

.97 
.97 

.664039 
.664371 

5.53 
5.53 

.335961 
.335629 

14 
13 

48 
49 
50 

.622682 
.622956 
.623229 

4.55 
4.57 
4.55 
4.55 

.957979 
.957921 
.957863 

.98 
.97 
.97 
.98 

.664703 
.665035 
.665366 

5.53 
5.53 
5.52 
5.53 

.335297 
.334965 
.334634 

12 
11 
10 

51 

52 

9.623502 
.623774 

4.53 

9.957804 
.957746 

.97 

QQ 

9.665698 
.666029 

5.52 

10.334302 
.333971 

9 

8 

53 
54 
55 
56 
57 

.624047 
.624319 
.624591 
.624863 
.625135 

4.55 
4.53 
4.53 
4.53 
4.53 

.957687 
.957628 
.957570 
.957M1 
.957452 

.yo 
.98 
.97 
.98 

.98 

.666360 
.666691 
.667021 
.667352 
.667682 

5.52 
5.52 
5.50 
5.52 
5.50 

.333640 
.333309 
.332979 
.332648 
.332318 

7 
6 
5 
4 
3 

58 
59 
60 

.625406 
.625677 
9.625948 

4.52 
4.52 
4.52 

.957393 
.957335 
9.957276 

.98 
.97 
.98 

.668013 
.668343 
9.668673 

5.52 
5.50 
5.50 

.331987 
.331657 
10.331327 

2 
1 
0 

' 

Cosine. 

D.  1*. 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

200 


65* 


COSINES,   TANGENTS,   AND   COTANGENTS. 


164* 


I 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

2 
3 

4 
5 
6 

7 

9.625948 
.626219 
.626490 
.626760 
.627030 
.627300 
.627570 
.627840 

4.52 
4.52 
4.50 
4.50 
4.50 
4.50 
4.50 
4  48 

9.957276 
.957217 
.957158 
.957099 
.957040 
.956981 
.956921 
.956862 

.98 
.98 
.98 
.98 
.98 
1.00 
.98 

QO 

1MJI5S073 
.669002 
.669332 
.669661 
.669991 
.670320 
.670649 
.670977 

5.48 
5.50 
5.48 
5.50 
5.48 
5.48 
5.47 

.880996 
.880668 
.880689 
.880009 

.829680 
.829851 
.829028 

60 
59 
58 
57 
56 
55 
54 
53 

8 

.628109 

440 

.956803 

QQ 

.671306 

5.48 

.826694 

B2 

9 

10 

.628378 
.628647 

4.48 
4.48 

.956744 
.956684 

1.00 

.98 

.671635 
.671963 

5.48 
5.47 
5.47 

.828865 

.826097 

51 
50 

11 
12 
13 
14 
15 
16 

9.628916 
.629185 
.629453 
.629721 
.629989 
.630257 

4.48 
4.47 
4.47 
4.47 
4.47 

4AK. 

9.956625 
.956566 
.956506 
.956447 
.956387 
.956327 

.98 
1.00 
.98 
1.00 
1.00 

no 

9.672291 
.672619 
.672947 
.673274 
.673602 
.673929 

5.47 
5.47 
5.45 
5.47 
5.45 

5Aff 

10.327709 
.327381 
.327053 
.826726 
.826896 
.326071 

49 
48 
47 
46 
45 
44 

17 
18 
19 

.630524 
.630792 
.631059 

4.47 
4.45 

44K 

.956268 
.956208 
.956148 

1.00 
1.00 

QQ 

.674257 
.674584 
.674911 

5.45 
5.45 

t  An 

.325743 
.325416 
.825060 

48 

42 
41 

20 

.631326 

4.45 

.956089 

1.00 

.675237 

5.45 

.324763 

40 

21 
22 
23 
24 
25 
26 

9.631593 
631859 
.632125 
.632392 
.632658 
.632923 

4.43 
4.43 
4.45 
4.43 
4.42 

9.956029 
.955969 
.955909 
.955849 
.955789 
.955729 

1.00 
1.00 
1.00 
1.00 
1.00 

9.675564 
.675890 
.676217 
.676543 
.676869 
.677194 

5  43 
5.45 
5.43 
5.43 
5.42 

10.324436 
.324110 
.828788 

.323457 
.323131 
4  .322806 

39 
38 
37 
36 
35 
34 

27 

28 

.633189 
.633454 

4.43 

4.42 

.955669 
.955609 

1.00 

.677520 
.677846 

5.43 

.822480 

,322154 

33 
32 

29 
30 

.633719 
.633984 

4.42 
4.42 
4.42 

.955548 
.955488 

1  00 
1.00 

.678171 
.678496 

5.42 
5.42 

.321829 
.321504 

31 
30 

31 
32 
33 
34 
35 
36 
37 

9.634249 
.634514 
.634778 
635042 
.635306 
.635570 
.635834 

4.42 
4.40 
4.40 
4.40 
4.40 
4.40 

4OQ 

9  955428 
.955368 
.955307 
.955247 
.955186 
.955126 
.955065 

1.00 
1.02 
1.00 
1.02 
1.00 
1.02 
1  00 

9.678821 
.  679146 
.679471 
.679795 
.680120 
.680444 
.680768 

5.42 
5  42 
5.40 
5.42 
5.40 
5.40 
5  40 

10.321179 
.820654 

.820529 

.320205 
.319880 
.319556 
.319232 

29 
28 
27 
26 
25 
24 
23 

38 
39 
40 

.636097 
636360 
.636623 

438 
4.38 
4  38 

.955005 
.954944 
.954883 

1.02 
1.02 
1  00 

.681092 
.681416 
.681740 

5.40 
5.40 
5.38 

.318908 
.318584 
.318260 

22 
21 
20 

41 
42 

43 
44 
45 

46 
47 
48 
49 
50 

9.636886 
637148 
.637411 
.637673 
.637935 
.638197 
.638458 
.638720 
.638981 
.639242 

4.37 
4.38 
4.37 
4.37 
4.37 
4.35 
4.37 
4.35 
4.35 
4  35 

9  954823 
.954762 
.954701 
.954640 
.954579 
.954518 
.954457 
.954396 
.954335 
.954274 

1  02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1  02 
1.02 
1  02 

9.682063 
.682387 
.682710 
.683033 
.683356 
.683679 
.684001 
.684324 
.684646 
.684968 

5.40 
5.38 
5.38 
5  38 
5.38 
5.37 
5.38 
5.37 
5.37 
5.37 

10  317937 
.317613 
.317290 
.816967 
.316644 
.316321 
.315999 
.315676 
.315354 
.315032 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.639503 
.639764 
.640024 
.640284 
.640544 
.640804 
.641064 
.641324 
.641583 
9.641842 

4.35 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.32 
4.32 

9.954213 
.954152 
.954090 
.954029 
.953968 
.953906 
.953845 
.953783 
.953722 
9.953660 

1.02 
1.03 
1.02 
1.02 
1  03 
1.02 
1.03 
1.02 
1.03 

9.685290 
.685612 
1  .685934 
686255 
.686577 
686898 
.687219 
.687540 
.687861 
9.688182 

5.37 
5.37 
5.35 
5.37 
5  35 
5.35 
5  35 
5.88 
5.35 

10  314710 
.314388 
814066 
313745 
.313423 
.313102 
.812781 
.312460 
.312139 
10.311818 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine.  !  D.  1'.  l|  Cotang. 

D.  1'.  I  Tang. 

115* 


201 


26° 


TABLE   XII. — LOGARITHMIC    SINES, 


153° 


' 

Sine. 

D.  I*. 

Cosine. 

D.l'. 

Tang. 

D.  1'. 

Cotang. 

> 

0 

1 

2 
3 
4 
5 
6 

9.641842 
.642101 
.642360 
.642618 
.642877 
.643135 
.643393 

4.32 
4.32 
4.30 
4.32 
4.30 
4.30 

4  Oft 

9.953660 
.953599 
.953537 
.953475 
.953413 
.953352 
.953290 

1.02 
1.03 
1.03 
1.03 
1.02 
1.03 
1  03 

9.688182 
.688502 
.688823 
.689143 
.689463 
.689783 
.690103 

5.33 
5.32 
5.33 
5.33 
5.33 
5.33 

5qq 

10.311818 
.311498 
.311177 
.310857 
.310537 
.310217 
.309897 

60 
59 
58 
57 
56 
55 
54 

8 
9 
10 

.643650 
.643908 
.644165 
.644423 

.60 

4.30 
4.28 
4.30 
4.28 

.953228 
.953166 
.953104 
.953042 

l.'OS 
1.03 
1.03 
1.03 

.690423 
.690742 
.691062 
.691381 

.00 

5.32 
5.33 
5.32 
5.32 

.309577 
.309258 
.308938 
,308619 

53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.644680 
.644936 
.645193 
.645450 
.645706 
.645962 
.646218 
.646474 
.646729 
.646984 

4.27 
4.28 
4.28 
4.27 
4.27 
4.27 
4.27 
4.25 
4.25 
4.27 

9.952980 
.952918 
.952855 
.952793 
.952731 
.952669 
.952606 
.952544 
.952481 
.952419 

1.03 
1.05 
1.03 
1.03 
1.03 
1.05 
1.03 
1.05 
1.03 
1.05 

9.691700 
.692019 
.6923';8 
.692656 
.692975 
.693293 
.693612 
.693930 
.694248 
.694566 

5.32 
5.32 
5.30 
5.32 
5.30 
5.32 
5.30 
5  30 
5.30 
5.28 

10.308300 
.307981 
.307662 
.307344 
.307025 
.306707 
.306388 
.306070 
.305752 
.305434 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 

9.647240 

4oq 

9.952356 

1  03 

9.694883 

10.305117 

39 

22 

.647494 

.6-J 

.952294 

|  .695201 

5  'oft 

.304799 

38 

23 

.647749 

4.25 

.952231 

1  05 

1  .695518 

.60 

K  qn 

.304482 

37 

24 
25 

.648004 
.648258 

4  .25 
4.23 

.952168 
.952106 

1^03 
1  05 

!  .695836 
|  .696153 

O.oU 

5.28 

K  Oft. 

.304164 
.303847 

36 
35 

26 

.648512 

A'  c*n 

.952043 

1  05 

!  .696470 

O  .60 

50ft 

.303530 

34 

27 
28 
29 
30 

.648766 
.649020 
.649274 
.649527 

4.6O 

4.23 
4.23 
4.22 
4.23 

.951980 
.951917 
.951854 
.951791 

lios 

1.05 
1.05 
1.05 

;  .696787 
i  .697103 
.697420 
.697736 

.60 

5.27 
5.28 
5.27 
5.28 

.303213 
.302897 
.302580 
.302264 

33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.649781 
.650034 
.650287 
.650539 
.650792 
.651044 
.651297 
.651549 
.651800 
.652052 

4.22 
4.22 
4.20 
4.22 
4.20 
4.22 
4.20 
4.18 
4.20 
4.20 

9.951728 
.951665 
.951602 
.951539 
.951476 
.951412 
.951349 
.951286 
.951222 
.951159 

1.05 
1  05 
1.05 
1  05 
1.07 
1  05 
105 
1.07 
1  05 
l!05 

1  9.698053 
.698369 
.698685 
.699001 
.699316 
.699632 
.699947 
.700263 
.700578 
.700893 

5.27 
5.27 
5.27 
5.25 
5.27 
5.25 
5.27 
5.25 
5.25 
5.25 

10.301947 
.301631 
.301315 
.300999 
.300684 
.300368 
.300053 
.299737 
.299422 
.299107 

29 

28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 

9.652304 
.652555 
.652806 
.653057 
653308 
.653558 

4.18 
4.18 
4.18 
4.18 
4.17 

9.951096 
.951032 
.950968 
.950905 
.950841 
.950778 

1.07 
1.07 
1.05 
1.07 
1.05 
1  07 

9.701208 
.701523 
.701837 
.702152 
.702466 
.702781 

5.25 
5.23 
5.25 
5.23 
5.25 

10.298792 

.298477 
.298163 
.297848 
.297534 
.297219 

19 
18 
17 
16 
15 
14 

47 
48 
49 
50 

.653808 
.654059 
.654309 
.654558 

4.17 
4.18 
4.17 
4.15 
4.17 

.950714 
.950650 
.950586 
.950522 

1  ."< 

1.07 
1.07 
1.07 
1.07 

.703095 
.703409 
.7037'22 
.704036 

5  .23 
5.23 
5.22 
5.23 
5.23 

.296905 
.296591 
.296278 
.295964 

13 
12 
11 
10 

51 

52 
53 
54 
55 
56 
57 
58 

9.634808 
.655058 
.655307 
.655556 
.655805 
.656054 
656302 
.656551 

4.17 
4.15 
4.15 
4.15 
4.15 
4.13 
4.15 

9.950458 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 
.950010 

1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
Ifift 

9.704350 
.704663 
.704976 
.705290 
.705603 
.705916 
.706228 
.706541 

5.22 
5.22 
5.23 
5.22 
5.22 
5.20 
5.22 

10.295650 
.295337 
.295024  i 
.294710 
.294397 
.294084 
.293772 
.293459 

9 
8 
7 
6 
5 
4 
3 
2 

59 
60 

.656799 
9.657047 

4.13 
4.13 

.949945 
9.949881 

.  Uo 

1.07 

.706854 
9.707166 

5.  '20 

.293146 

10.292834 

1 
0 

'  1 

Cosine. 

D.l\ 

Sine.   D.  1".   Cotang. 

D.r.  I 

Tang. 

it 

116* 


202 


63* 


27° 


COSINES,   TANGENTS,   AND   COTANGENTS.  162° 


• 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

9.657047 
.657295 
.657542 
.657790 

4.13  i 
4.12 
4.13 

9.949881 
.949816 
.949762 
.949688 

1.08 
1.07 
1.07  ! 

9.707166 
.707478 
.707790 
.708102 

5.20 
5.20 
5.20 

.898210 
.891898 

60 
59 
58 
57 

4 
5 
6 

.658037 
.658284 
.658531 

4.12 
4.12 
410 

.9496213 
.949558 
.949494 

1.08 
1.07 

1  Oft 

.708414 
.708788 

.709037 

5.80 

5.18 
Son 

.891586 
.890968 

56 
65 

54 

.658778 

.949429 

.709349 

518 

.200861 

58 

8 
9 
10 

.659025 
.659271 
659517 

4.10 
4.10 
4.10 

.949364 
.949300 
.949235 

1.07 
1.08 
1.08 

.709660 
.709971 
.710282 

5.18 
5.18 
6.18 

.890640 
.890089 

.289718 

51 

51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.659763 
.660009 
.660255 
.660501 
660746 
.660991 
.661236 
.661481 
.661726 
.66197'0 

4.10 
4.10 
4.10 
4.08 
4.08 
4.08 
4.08 
4.08 
4.07 
4  07 

9.949170 
.949105 
.949040 
.948975 
.948910 
.948845 
,948780 
.948715 
.948650 
.948584 

1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.10 
1  08 

9.710593 
.710904 
.711215 
.711525 
.711836 
.712146 
.712456 
.712766 
.713076 
.713386 

5.18 
5.18 
5.17 
5.18 
5.17 
5.17 
5.17 
5.17 
5.17 
5.17 

10.289407 
.889096 
.888785 
.888475 
.888164 
.887854 
.887544 

.286614 

40 
H 
47 
46 
45 
44 
43 
42 
41 
40 

21 

9.662214 

4AQ 

9.948519 

1  Aft 

9.713696 

5  15 

10.286304 

30 

22 
23 

24 
25 

26 
27 
28 
29 
30 

.662459 
.662703 
.  662946 
.663190 
.663433 
.663677 
.663920 
.664163 
.664406 

4.07 
4.05 
4.07 
4.05 
4.07 
4.05 
4.05 
4.05 
4  03 

.948454 
.948388 
.948323 
.948257 
.948192 
.948126 
.948060 
.947995 
.947929 

1.10 
1.08 
1.10 
1.08 
1.10 
1.10 
1.08 
1.10 
1  10 

.714005 
.714314 
.714624 
.7149313 
.715242 
.715551 
.715860 
.716168 
.716477 

5.15 
5.17 
5.15 
5.15 
5.15 
5.15 
5.13 
5.15 
5.13 

.886996 
.885686 

.285376 
.885067 

.284758 
.284449 
.2K41  !0 
.283832 
.88800 

as 

37 
36 
,35 
34 
33 
32 
31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
40 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.664648 
.664891 
.665133 
.665375 
.665617 
.665859 
.666100 
.666342 
.666583 
.666824 

9.667065 
.667305 
.667546 
.667786 
.668027 
.668267 
.668506 
.668746 
.668986 
.669225 
'  9.660464 
669703 
(509912 
670181 
670419 
.670(558 
670896 
.671134 
.671372 
9.671609 

4.05 
4.03 
4.03 
4.03 
4.03 
4.02 
4.03 
4.02 
4.02 
4.02 

4.00 
4.02 
4.00 
4.02 
4.00 
3.98 
4.00 
4.00 
3.98 
3.98 

3.98 
3.98 
3.98 
3.97 
3.98 
3.97 
3.97 
3.97 
3.95 

9.947863 
.947797 
.947731 
.947665 
.947600 
.947533 
.947467 
.947401 
.947335 
.947269 

9.947203 
.947136 
.947070 
.947004 
.946937 
.946871 
946804 
.946738 
.946671 
.946604 

9.946538 
.946471 
.946404 

946337 
.946270 
946203 
.946136 
.946069 
.946002 
9.945935 

1  10 
1.10 

1.10 
1.08 
1.12 
1.10 
1.10 
1.10 
1.10 
1.10 

1.12 
1.10 
1.10 
1.12 
1.10 
1.12 
1.10 
1.12 
1.12 
1.10 

1.12 
1.12 
1.12 
1.12 
1.12 
1.18 
1.18 
1.12 
1.12 

9.716785 
.717093 
.717401 
.717709 
.718017 
.718325 
.718633 
.718940 
.719248 
.719555 

9.719862 
.720169 
.720476 
.720783 
.721089 
.721396 
.721702 
.722009 
.722315 
.788681 

9.722027 
723232 

.788844 
7*4141 
784464 
78476! 
785060 

5  13 
5.13 
5.13 
5.13 
5.13 
5.13 
5.12 
5.13 
5.12 
5.12 

5.12 
5.12 
5.12 
5.10 
5.12 
5.10 
5.12 
5.10 
5.10 
5.10 

5.08 
5.10 
5.10 

5.10 

5.  OS 
5.  OS 
5.07 

10.283215 
.888907 
.888599 
.888891 
881988 
.281675 
.2811367 
.881060 
880758 
.280446 

10.280138 
879881 

.279217 
.278911 
[878601 
.878898 

• 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
18 
11 
10 

9 
H 
7 
6 
5 
4 

i 
9 

' 

Cosine. 

D.  1". 

Sine,  i  D.  1".   Cotang 

i).  r. 

Tang.  |  ' 
~^fi& 

117° 


203 


28° 


TABLE   XII. — LOGAKITHMIC   SINES, 


161° 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

9.671609 
.671847 

3.97 

9.945935 
.945868 

1.12 

9.725674 

.725979 

5.08 

10.274326 
.274021 

60 
59 

2 
3 

.672084 
.672321 

3.95 
3.95 

.945800 
.945733 

1.12 

.726284 
.726588 

5.08 
5.07 

.273716 
.273412 

58 
57 

4 

.672558 

3.95 

.945666 

1  .12 

.726892 

5.07 

.273108 

56 

5 
6 

7 

.672795 
.673032 
.673268 

3.95 
3.95 
3.93 

.945598 
.945531 
.945464 

1.13 
1.12 

1.12 

.727197 
.727501 

.727805 

5.05 
5.07 
5.07 

.272803 
.272499 
.272195 

55 
54 
53 

8 

.673505 

3.95 

.945396 

1  .13 

.728109 

5.07 

.271891 

52 

9 
10 

.673741 
.673977 

3.93 
3.93 
3.93 

.945328 
.945261 

1  .13 
1.12 
1.13 

.728412 
.728716 

5.05 
5.07 
5.07 

.271588 
.271284 

51 
50 

11 
12 

9.674213 
.674448 

3.92 

9.945193 
.945125 

1.13 

9.729020 
.729323 

5.05 

10.270980 
.270677 

49 
48 

13 

.674684 

3.93 

.945058 

1.12 

.729626 

5.05 

.270374 

47 

14 

.674919 

3.92 

.944990 

1.13 

.729929 

5.05 

.270071 

46 

15 

.675155 

3.93 

.944922 

1.13 

.730233 

5.07 

.269767 

45 

16 

.675390 

3.92 

.944854 

1.13 

.730535 

5.03 

.269465 

44 

17 

.675624 

3.90 

.944786 

1.13 

.730838 

5.05 

.269162 

43 

18 

.675859 

3.92 

.944718 

1.13 

.731141 

5.05 

.268859 

42 

19 
20 

.676094 
.676328 

3.92 
3.90 

.944650 
.944582 

1.13 
1.13 

.731444 
.731746 

5.05 
5.03 

.268556 
.268254 

41 
40 

3.90 

1.13 

5.03 

21 

9.676562 

9.944514 

9.732048 

10.267952 

39 

22 

.676796 

3.90 

.944446 

1  .13 

.732351 

5.05 

.267649 

38 

23 

.677030 

3.90 

.944377 

1  .15 

.732653 

5.03 

.267347 

37 

24 
25 

26 
27 

28 
29 
30 

.677264 
.677498 
.677731 
.677964 
.678197 
.678430 
.678663 

3.90 
3.90 
3.88 
3.88 
3.88 
3.88 
3.88 
3.87 

.944309 
.944241 
.944172 
.944104 
.944036 
.943967 
.943899 

1.13 
1.13 
1.15 
1.13 
1.13 
1.15 
1.13 
1.15 

.732955 
.733257 
.733558 
.733860 
.734162 
.734463 
.734764 

5.03 
5.03 
5.02 
5.03 
5.03 
5.02 
5.02 
5.03 

.267045 
.266743 
.266442 
.266140 
.265838 
.265537 
.265236 

36 
35 
34 
33 
32 
31 
30 

31 
32 

9.678895 
.679128 

3.88 

9.943830 
.943761 

1.15 

9.735066 
.735367 

5.02 

10.264934 
.264633 

29 

28 

33 
34 
35 
36 

.679360 
.679592 
.679824 
.680056 

3.87 
3.87 
3.87 
3.87 

.943693 
.943624 
.943555 
.943486 

1.13 
1.15 
1.15 
1.15 

.735668 
.735969 
.736269 
.736570 

5.02 
5.02 
5.00 
5.02 

.264332 
.264031 
.263731 
.263430 

27 
26 
25 
24 

37 

38 

.680288 
.680519 

3.87 
3.85 

.943417 
.943348 

1.15 
1.15 

.736870 
.737171 

5.00 
5.02 

.263130 

.262829 

23 

22 

39 
40 

.680750 
.680982 

3.85 
3.87 
3.85 

.943279 
.943210 

1.15 
1.15 
1.15 

.737471 
.737771 

5.00 
5.00 
5.00 

.262529 
.262229 

21 
20 

41 
42 
43 
44 

9.681213 
.681443 
.681674 
.681905 

3.83 

3.85 
3.85 

9.943141 
.943072 
.943003 
.942934 

1.15 
1.15 
1.15 

9.738671 
.738371 
.738671 
.738971 

5.00 
5.00 
5.00 

10.261929 
.261629 
.261329 
.261029 

19 

18 
17 
16 

45 
46 

47 
48 
49 
50 

.682135 
.682365 
.682595 
.682825 
.683055 
.683284 

3.83 
3.83 
3.83 
3.83 
3.83 
3.82 
3.83 

.942864 
.942795 
.942726 
.942656 
.942587 
.942517 

1.17 
1.15 
1.15 
1.17 
1.15 
1.17 
1.15 

.739271 
.739570 
.739870 
.740169 
.740468 
.740767 

5.00 
4.98 
5.00 

4.98 
4.98 
4.98 
4.98 

.260729 
.260430 
.260130 
.259831 
.259532 
.259233 

15 

14 
13 
12 
11 
10 

51 

52 

9.683514 

.683743 

3.82 

9.942448 
.942378 

1.17 

9.741066 
.741365 

4.98 

10.258934 
.258635 

9 

8 

53 
54 
55 

.683972 
.684201 
.684430 

3.82 
3.82 
3.82 

.942308 
.942239 
.942169 

1.17 
1.15 
1.17 

.741664 
.741962 
.742261 

4.98 
4.97 
4.98 

.258336 
.258038 
.257739 

7 
6 
5 

56 
57 
58 
59 
60 

.684658 
.684887 
.685115 
.685343 
9.685571 

3.80 
3.82 
3.80 
3.80 
3.80 

.942099 
.942029 
.941959 
.941889 
9.941819 

1.17 
1.17 
1.17 
1.17 
1.17 

.742559 
.742858 
I  .743156 
.743454 
9.743752 

4.97 
4.98 
4.97 
4.97 
4.97 

.257441 
.257142 
.256844 
.256546 
10.256248 

4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

118* 


204 


61- 


COSINES,    TANGENTS,    AND    COTANGENTS.  150° 


/ 

Sine. 

D.  1". 

Cosine. 

D.  1*. 

Tang. 

D.  r. 

Cotang. 

• 

0 

9.<KV>71 

3Qf\ 

9.941R19 

11*"* 

40" 

GO' 

1 

2 

!  685799 
.686027 

3.80 

0  r-Q 

.941749 
.941679 

1.17 
1  17 

.744050 
744348 

4  Mr 

5H 

3 
4 
5 

.686254 
.686482 

.680709 

3.  SO 
3.78 

0  r~Q 

.941609 

.941539 
.941469 

1.17  i 
1.17 
1  18 

.744645 

.744943 
.745240 

4.87 

4.95 
4  97 

.255057 
.254760 

57 

56 
55 

6 

7 

.686936 
.687163 

3.78 

.941398 
.941328 

1.17 

1  17 

.  74*5885 

4.95 

.254462 

54 
58 

8 
9 

10 

.687389 
.687616 
.687843 

3.78 
3.78 
3.77  1 

.941258 
.941187 

.941117 

1.18 
1.17 

1.18 

.74ol32 
.746489 
.746726 

4.95 
4.95 
4.95 

. 

92 

51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.  688069 
.688295 
.688521 
.688747 
.688972 
.689198 
.689423 
.689648 
.689873 
.690098 

3.77 
3.77 
3.77 
3.75 
3.77 
3  .  75 
3.75 
3.75  , 
3.75  | 
3.75 

9.941040 
.940975 
.940905 
.940834 
.940763 
.940693 
.940022 
.940551 
.940480 
.940409 

1.18 
1.17 
1.18 
1.18 
1.17 
1.18 
1.18 
1.18 
1.18 
1.18 

9.747023 
.747319 
.747010 
.747913 
.748209 
.74&505 
.748801 
.749097 
.749393 
.749689 

4.93 
4.95 
4.95 
4.93 
4.93 
4.93 
4.93 
4.93 
4.03 
4.93 

259087 
.251791 
.251495 

.251199 
.250903 
.250807 

.250311 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 

9.690323 
.690548 
.690772 
.690996 
.691220 
.691444 
.691668 
.691892 

3.75 
3.73 

3.73  I 
3.73  | 
3.73  1 
3.73  j 
3.73  I 

9.940338 
.940207 
.940196 
.940125 
.940054 
.939982 
.939911 
.939840 

1.18 
1.18 
1.18 
1.18 
1.20 
1.18 
1.18 

9.749985 
.750281 
.750576 
.750872 
.751167 
.751462 
.751757 
.752052 

4.93 
4.92 
4.93 
4.92 
4.92 
4.92 
4.92 

4Q9 

10  250015 
.249719 
.248424 
.249128 
.248833 
.248588 
.248243 
.247948 

39 
38 

86 

35 
34 
33 
32 

29 
30 

.692115 
.692339 

3.72  1 
3.73 
3  72 

.939768 
.939697 

1.18 
1.20 

.752347 
.752642 

4.92 
4.92 

.247653 
.247358 

31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.692562 
.692785 
.693008 
.693231 
.693453 
.693676 
.693898 
.694120 
.694342 
.694564 

3.72 
3.72 
3.72 
3.70 
3.72 
3.70 
3.70 
3.70 
3.70 
3  70 

9.939625 
.939554 
.939482 
.939410 
.939339 
.939207 
.939195 
.939123 
.939052 
.938980 

1.18 
1.20 
1  20 
1.18 
1.20 
1.20 
1.20 
1.18 
1.20 
1.20 

9.752937 
.753231 
.753526 
.753820 
.754115 
.754409 
.754703 
.754997 
.755291 
.755585 

4.90 
4  92 
4.90 
4.92 
4.90 
4.90 
4.90 
4.90 
4.90 
4.88 

10.247063 
.246709 
.240474 
.246180 
.245885 
.245591 
245297 
.245003 
.244709 
.244415 

29 
28 
27 
26 
29 
xl4 
23 
22 
21 
20 

41 
42 

43 
44 
45 
46 
47 
48 
49 
50 

9.694786 
.695007 
.695229 
.695450 
.695671 
.695892 
.696113 
.696334 
.696554 
.090775 

3.68 
3.70 
3.68 
3  68 
3.68 
3.68 
3.68 
3.  (57 
3.68 
3  07 

9.938908 
.938836 
.938763 
.938691 
.938619 
.938547 
.938475 
.938402 
.938330 
.938258 

1.20 
1.22 
1.20 
1.20 
1.20 
1.20 
1.22 
1.20 
1.20 
1  °2 

9.755878 
.756172 
.756465 
.756759 
.757052 
.757345 
.757638 
.757931 
.758224 
.758517 

4.90 
4.88 
4.90 
4.88 
4.88 
4.88 
4.88 
4.88 
4.88 
4.88 

10.244122 
.243535 
.242948 
.242862 

'.  241  776 
.241483 

19 
18 
17 
16 
15 
14 
18 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.696995 
.697215 
.697435 
.697654 
.697874 
.698094 
.698313 
.698532 
.698751 
9.698970 

3.07 
3.67 
3.65 
3.67 
3.67 
3.65 
3.65 
3.65 
3.65 

9.938185 
.938113 
.938040 
.937967 
.937895 
.937822 
.937749 
.937676 
.937601 
9.937531 

1.20 
1.22 
1.22 
1.20 
1.22 
1.22 
1.22 
1.20 
1.22 

9.758810 
.759102 
.759395 
.759687 
759979 
.760272 
.760564 
.760856 
.761148 
8.761489 

4.87 
4.KS 
4.87 
4.87 
4.88 
4.87 
4  ^7 

4.85 

10.241190 
.240898 
840605 
240818 

.240021 

.888808 

10.238561 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.r. 

Sine. 

D.  1'. 

Cotang.  D.  1'.   Tang. 

119° 


205 


30° 


TABLE   XII. — LOGARITHMIC    SINES, 


149° 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.698970 
.699189 

3.65 

3  no 

9.937531 
.937458 

1.22 

9.761439 
.761731 

4.87 

10.238561 

.238269 

60 
59 

2 
3 

.699407 
.699626 

.  Do 

3.65 

3  CO 

.937385 
.937312 

.22  | 
.22 

00 

.762023 
.762314 

4.87 
4.85 

.237977 
.237686 

58 
57 

4 

.699844 

.DO 
3K°. 

.937238 

.9*9 

.762606 

4.87 

.237394 

56 

5 

.700062 

.Do 

3  63 

.937165 

.22 

.762897 

4.85 

.237103 

55 

6 
7 
8 
9 

.700280 
.700498 
.700716 

3^63 
3.63 
3.62 
3  63 

.937092 
.937019 
.936946 
.936872 

!22 
.22 
.88 
oo 

.763188 
.763479 
.763770 
.764061 

4.85 
4.85 
4.85 
4.85 

.236812 
.236521 
.236230 
.235939 

54 
53 

52 
51 

10 

!  701  151 

3^62 

.936799 

'.23 

.764352 

4.85 
4.85 

.235648 

50 

11 

9.701368 

3  62 

9.936725 

9.764643 

10.235357 

49 

12 

.7J1585 

.936652 

go 

.764933 

4.83 

.235067 

48 

13 

.701802 

3  62 

.936578 

.  4iO 

.765224 

4.85 

.234776 

47 

14 
15 
16 

.702019 
.702236 
.702452 

3^62 
3.60 
3  62 

.936505 
.936431 
.936357 

^23 
.23 
22 

.765514 

.765805 
.766095 

4.83 

4.85 
4.83 

4  DO 

.234486 
.234195 
.233905 

46 
45 
44 

17 

18 

.702669 

.702885 

3^60 
3  60 

.936284 
.936210 

^23 
23 

.766385 
.766675 

OO 

4.83 

400 

.233615 
.233325 

43 

42 

19 

.703101 

3  60 

.936136 

oq 

.766965 

.00 

.233035 

41 

20 

.703317 

3^60 

.936062 

.6<J 

.767255 

4.83 
4.83 

.232745 

40 

21 
22 

9.703533 
.703749 

3.60 
3  58 

9.935988 
.935914 

1.23 

9.767545 
.767834 

4.82 

4   DO 

10.232455 
.232166 

39 

38 

23 
24 
25 
26 

.703964 
.704179 
.704395 
.704610 

3^58 
3.60 
3.58 
3  58 

.935840 
.935766 
.935692 
.935618 

1.23 
1.23 
1.23 

.768124 
.768414 
.768703 
.768992 

.OO 

4.83 
4.82 
4.82 

.231876 
.231586 
.231297 
.231008 

37 

36 
35 
34 

27 

28 

.704825 
.705040 

3^58 
3  57 

.935543 
.935469 

1^23 

1  9°, 

.769281 
.769571 

4.82 
4.83 

.230719 
.230429 

33 
32 

29 
30 

.705254 
.705469 

3^58 
3.57 

.935395 
.935320 

1  ,/4O 

1.25 
1.23 

.769860 
.770148 

4.82 
4.80 
4.82 

.230140 
.229852 

31 
30 

31 

9-705683 

3  58 

9.935246 

9.770437 

4  DO 

10.229563 

29 

32 
33 
34 
35 
36 

.705898 
.706112 
.706326 
.706539 
.706753 

3^57 
3.57 
3.55 
3.57 

3KTf 

.935171 
.935097 
.935022 
.934948 
.934873 

l'.23 
1.25 
1.23 
1.25 

.770726 
.771015 
.771303 
.771592 

.771880 

.O-* 

4.82 
4.80 
4.82 
4.80 

.229274 

.228985 
.228697 
.228408 
.228120 

28 
27 
26 
25 
24 

37 

.706967 

.Of 

3tpr 

.934798 

i  '25 

.772168 

4.80 

.227832 

23 

38 
39 

.707180 
.707393 

.OO 

3-55 
3  55 

.934723 
.934649 

1^23 

.772457 

.772745 

4.82 
4.80 

4  on 

227543 
!  227255 

22 
21 

40 

.707606 

3^55 

.934574 

1.'25 

.773033 

.OU 

4.80 

.226967 

20 

41 
42 
43 

9.707819 
.708032 
.708245 

3.55 
3.55 
3  55 

9.934499 
.934424 
.934349 

1.25 
1.25 

1OK 

9.773321 

.773608 

.773896 

4.78 
4.80 

10.226679 
.226392 
.226104 

19 
18 
17 

44 
45 

.708458 
.708670 

3^53 
3  53 

.934274 
.934199 

.^D 

1.25 

197 

.774184 
.774471 

4>8 

4  DA 

.225816 
.225529 

16 
15 

46 
47 
48 
49 
50 

.708882 
.709094 
.709306 
.709518 
.709730 

3^53 
3.53 
3.53 
3.53 
3.52 

.934123 
.934048 
.933973 
.933898 
.933822 

.*! 

1.25 
1.25 
1.25 
1.27 
1.25 

.774759 
.775046 
.775333 
.775621 
.775908 

.OU 

4.78 
4.78 
4.80 
4.78 
4.78 

.225241 
.224954 

.224667 
.224379 
.224092 

14 
13 
12 
11 
10 

51 
52 
53 
54 
55 

9.709941 
.710153 
.710364 
.710575 
.710786 

3.53 
3.52 
3.52 
3.52 

9.933747 
.933671 
.933596 
.933520 
933445 

1.27 
1.25 
1.27 
1.25 

9.776195 

.776482 
.776768 
.777055 
.777342 

4.78 
4.77 
4.78 
4.78 

477 

10.223805 
.223518 
.223232 
.222945 

.222658 

9 
8 
7 
6 
5 

56 
57 
58 
59 
60 

.710997 
.711208 
.711419 
.711629 
9.711839 

3.52 
3.52 
3.52 
3.50 
3.50 

.933369 
.933293 
.933217 
.933141 
9.933066 

1.27 
1.27 
1.27 
1.27 
1.25 

.777628 
.777915 

.778201 
.778488 
9.778774 

.  I  ( 

4.78 
4.77 
4.78 
4.77 

.222372 

.222085 
.221799 
.221512 
10.221226 

4 
3 
2 
1 
0 

' 

Cosine. 

D.  1".  | 

I   Sine. 

D.  r. 

Cotang. 

D.  1*. 

Tang. 

' 

120° 


206 


59' 


COSINES,    TANGENTS,    AND    COTANGENTS.  148° 


/ 

$ine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

/ 

0 

9.711839 

3  to 

9.033066 

197 

9.778774 

4r»7 

60 

1 

2 
3 

4 
5 
6 

.712050 
.712260 
.712469 
.712679 
712889 
713098 
713308 

3.50 
3.48 
3.50 
3.50 
3.48 
3.50 

3AO 

.988990 
.682914 

.932838 
.932762 
.U*J»>«5 
.932609 
.932533 

1.27 
1.27 
1.27 
1.28  : 
1.27 
1.27  : 

1O7     i 

.779080 

.779346 
.779883 
.779918 

.780203 
.780489 
.780775 

4.77 
4.77 
4.77 
4.75 
4.77 
4.77 

.220940 

.220088 

.219797 
.219611 

.219225 

59 
58 
57 
56 
55 
54 
53 

8 
9 
10 

713517 
713726 
.713935 

3.48 
3.48 
3.48 

.932457 
.932380 
.932304 

1.28  ! 
1.27  I 
1.27  ! 

.781060 
.781346 
.781631 

4.77 
4.75 
4.75 

.218940 
.218864 

.218369 

• 

52 
51 
50 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 

9.714144 
.714352 
.714561 
.714769 
.714978 
.715186 
.715394 
.715602 
.715809 
.716017 

3.47 
3.48 
3.47 
3.48 
3.47 
3.47 
3.47 
3.45 
3.47 
3.45 

9.932228 
.932151 
.932075 
.931998 
.931921 
.931845 
.931768 
.931691 
.931614 
.931537 

1.28  ! 
1.27 
1.28 
1.28 
1.27 
1.28 
1.28 
1.28 
1.28 
1.28 

9.781916 
.782201 
.782486 
.782771 
.783056 
.783341 
.783626 
.783910 
.784195 
.784479 

4.75 
4.75 
4.75 
4.75 
4.75 
4.75 
4.73 
4.75 
4.73 
4.75 

10.218084 
.217799 
.217514 
.217229 
.218944 
.216659 
.216374 
.216000 
.215805 
.215521 

49 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.716224 
.716432 
.7J6639 
.716846 
.717053 
.717259 
.717466 
.717673 
.717879 
.718085 

3.47 
3.45 
3.45 
3.45 
3.43 
3.45 
3.45 
3.43 
3.43 
3  43 

9.931460 
•  .931383 
.93130(5 
.931229 
.931152 
.931075 
.930998 
.930921 
.930843 
.930766 

1.28 
1.28 
1.28 
1.28 
1.28  [ 
1.28 
1.28 
1.30  ! 
1.28 
1.30 

9.784764 
.785048 
.785332 
.785616 
.785900 
.786184 
.786468 
.786752 
.787036 
.787319 

4.73 
4.73 
4.73 
4.73 
4.73 
4.73 
4.73 
4.73 
4.72 
4.73 

10.215236 
.214952 
.214668 
.214384 
.214100 
.213816 
.213532 
.213248 
.212984 
.212881 

39 
38 
37 
36 
35 
34 
33 
82 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.718291 
.718497 
.718703 
.718909 
.719114 
.719320 
.719525 
.719730 
.719935 
.720140 

3.43 
3.43 
3.43 
3.42 
3.43 
3.42 
3.42 
3.42 
3.42 
3  42 

9.930688 
.930611 
.930533 
.930456 
.930378 
.930300 
.930223 
.930145 
.930067 
.929989 

1.28 
1.30 
1.28 
1.30 
1.30 
1.28 
1.30 
1.30 
1.30 
1  30 

9.787603 
.787886 
.788170 
.788453 
.788736 
.789019 
.789302 
.789585 
.789868 
.790151 

4.72 
4.73 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 

10.212397 
.212114 

.211830 
.211547 
.211264 
.210981 
.210698 
.210415 
.210132 
.209849 

29 
28 
27 
26 
25 
24 
23 
22 
81 
80 

41 

42 
43 
44 
45 
46 
47 
48 
49 
50 

9.720345 
.720549 
.720754 
.720958 
.721162 
.721366 
.721570 
.721774 
.721978 
.722181 

3.40 
3.42 
3.40 
3.40 
3.40 
3.40 
3.40 
3.40 
3.38 

3A(\ 

9.929911 
.929833 
.929755 
.929677 
.929599 
.929521 
.929442 
.929364 
.929286 
.929207 

1.30 
1.30 
1.30 
1.30 
1.30 
1.32 
1.30 
1  30 
1.32 
1  30 

9.790434 
.790716 
.790999 
.791281 
.791563 
.791846 
.792128 
.792410 
.792692 
.792974 

4.70 
4.72 
4.70 
4.70 

4'.70 
4.70 
4.70 
4.70 
4.70 

10.209566 
.20928* 

.209001 

1208487 

.208154 
.207872 

.207308 
.207088 

19 

18 

18 

15 
14 
13 
18 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.722385 
.722588 
.722791 
,722994 
.723197 
.723400 
.723603 
.723805 
.724007 
9.724210 

3.38 
8.88 

3.38 
3.38 
3.38 
3.38 
3.37 
3.37 
3.38 

9.929129 
.929050 
.928972 
,928898 
.988815 
.928736 
.928657 
.928578 
.02S4W 
9.928420 

1.32 
1.30 
1.32 
1.30 
1.32 
1.32 
1.32 

lisa 

1.32 

9.79325f 
.798538 
.79881! 
.794101 
.7943*3 
79488 
.794914 
.  79582-3 
.796608 
9.7957K. 

4.70 
4.68 
4.70 
4.70 
4.68 
4.70 
4.68 
4.68 
4.68 

10.206744 

.208181 
.206890 
.205817 
;  206888 
.206064 

.804498 

9 

8 

6 
5 

4 
3 

1 
0 

/ 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang 

D.  1'. 

Tang. 

68* 

121 


207 


32° 


TABLE   XII. — LOGARITHMIC   SINES, 


147° 


1 

' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

0.1*. 

Cotang. 

' 

0 

1 

9.724210 

.724412 

3.37 

39.7 

9.928420 

.928342 

1.30 

9.795789 
.796070 

4.68 
4  AS 

10.204211 
.203930 

CO 
50 

2 

.724614 

•  Ol 

.928263 

?*£» 

.796351 

.Oo 

4  Aft 

.203649 

58 

3 

.724816 

3.37 

39.K 

.938183 

1  99 

.796632 

.Do 
4  Aft 

.203368 

57 

4 

.725017 

.OO 
39.7 

.928104 

Iqo 

.796913 

.Do 
4  Aft 

.203087 

56 

5 

.725219 

.of 

.928025 

,«£• 

.797194 

.Do 
4A7 

.202806 

55 

6 

.725420 

3.35 

39.7 

.927946 

1  99 

.797474 

.Of 

4  Aft 

.202526 

54 

7 

.725622 

.ot 

q  qK 

.927867 

1  33 

.797755 

.  Do 
4  gg 

.202245 

53 

8 

.725823 

O.OO 
39.K 

.927787 

.798036 

.201064 

52 

9 

.726024 

.OO 
3  OK 

.927708 

1  99 

.798316 

4A1"* 

.201684 

51 

10 

.726225 

.OO 

3.35 

.927629 

lias 

.798596 

.Oi 

4.68 

.201404 

50 

11 

9.726426 

9.927549 

1  9.9 

9.798877 

4A1** 

10.201123 

40 

12 

.726626 

o.oo 

3qK 

.927470 

Iqq 

.799157 

.  Ol 
4A7 

.200843  ;  48 

13 

.726827 

.  OO 

.927390 

.00 

Iqq 

.799437 

.VI 

4A7 

.200563 

47 

14 
15 

.727027 
.727228 

3.33 
3.35 

.927310 
.927231 

.00 

1.32 

.799717 
.799997 

-0< 

4.67 

.200283 
.200003 

46 
45 

16 
17 
18 
19 
20 

.727428 
.727628 
.727828 
.728027 

.728227 

3.33 
3  33 
3.33 
3.32 
3.33 
3.33 

.927151 
.927071 
.926991 
.926911 
.926831 

li33 
1.33 
1.33 
1.33 
1.33 

.800277 
.800557 
.800836 
.801116 
.801396 

4i67 
4.65 
4.67 
4.67 
4.65 

.199723 
.199443 
.190164 

.108884 
.108604 

4-1 

43 
42 
41 
40 

21 

fl.  728427 

9.926751 

1  9.9. 

9.801675 

10.198325 

30 

22 
23 
24 

,728626 
.728825 
.729024 

3^32 
3.32 

.926671 
.926591 
.926511 

1  .  OO 

1.33 
1.33 

1  99 

.801955 
.802234 
.802513 

4^65 
4.65 

.198045 
.107766 
.107487 

38 
37 
36 

25 
26 
27 

.729223 
.729422 
.729621 

3.32 
3.32 
3.32 

3qo 

.926431 
.926351 
,  926270 

1  .OO 

1.33 
1.35 

Iqq 

.802792 
.803072 
.803351 

4.65 
4.67 
4.65 

40-r 

.107208 
.106028 
.106640 

35 
34 
33 

28 
29 
30 

.729820 
.730018 
.730217 

.CM 

3.30 
3.32 
3.30 

.926190 
.926110 
.926029 

.  OO 

1.33 
1.35 
1.33 

.803630 
.803909 
.804187 

.  OO 

4.65 
4.63 
4.65 

.106370 
.106091 
.195813 

32 
31 
30 

31 

9  730415 

39.fl 

9.925949 

Q- 

9.804466 

10.195534 

29 

32 

.730613 

.  OU 

.925868 

1  ,OO 

.804745 

4.  60 

.105255 

28 

33 
34 
35 
36 
37 
38 
39 
40 

.730811 
.731009 
.731206 
.73J404 
.731602 
.731799 
.731996 
.732193 

3.30 
3.30 
3.28 
3.30 
3.30 
3.28 
3.28 
3.28 
3.28 

.925788 
.925707 
.925626 
.925545 
.925465 
.925384 
.925303 
.925222 

1.33 
1.35 
1.35 
1.35 
1.33 
1.35 
1.35 
1.35 
1.35 

.805023 
.805302 
.805580 
.805859 
.806137 
.806415 
.806693 
.806971 

4.63 
4  65 
4.63 
4.65 
4  63 
4.63 
4.63 
4.63 
4.63 

.194977 
.194698 
.194420 
.194141 
.193863 
.103585 
.103307 
.193029 

27 
26 
25 
24 
23 
22 
21 
20 

41 
42 

9.732390 
.7'32587 

3.28 

39ft 

9.925141 
.925060 

1.35 

9.807249 
.807527 

4.63 

4A9. 

10.102751 
.102473 

19 

IS 

43 
44 
45 

46 
47 
48 

.732784 
.732980 
.733177 
.733373 
.733569 
.733765 

.  *o 

3.27 
3.28 
3.27 
3.27 
3.27 

.924979 
.924897 
.924816 
.924735 
.924654 
.924572 

1:37 
1.35 
1.35 
1.35 
1.37 

IqK 

.807805 
.808083 
.808361 
.808638 
.808916 
.809193 

.OO 

4.63 
4.63 
4.62 
4.63 
4.62 

4A9. 

.102195 
.191017 
.101639 
.191362 
.191084 
.190807 

17 
16 
15 
14 
13 
12 

49 
50 

.733961 
.734157 

3^27 
3.27 

.924491 
.924409 

.OD 

1.37 
1.35 

.809471 
.809748 

.Oo 

4.62 
4.62 

.100520 
.190252 

11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.734353 
.734549 
.734744 
.734939 
.735135 
.735330 
.735525 
.735719 
.735914 
9.736109 

3.27 
3.25 
3.25 
3.27 
3.25 
3.25 
3.23 
3.25 
3.25 

9.924328 
.924246 
.924164 
.924083 
.924001 
.923919 
.923837 
.023755 
.923673 
9.023501 

1.37 
1.37 
1.35 
1.37 
1.37 
1.37 
1  37 
1.37 
1.37 

9.  SI  0025 
.810302 
.810580 
.810857 
.811134 
.811410 
.811687 
.811964 
.812241 
9.812517 

4.62 
4.63 
4.62 
4.62 
4.60 
4.62 
4.62 
4.62 
4.60 

10.1  ROOTS 
.  180608 
.189420 
.180143 
.188866 
.188500 
.188313 
.188036 
.187759 
10.187483 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

'   Cosine. 

D.  1".  i  Sine.   D.  1'.  i  Cotang.  D.  1". 

Tang. 

' 

1230 


208 


57- 


COSINES,,    TANGENTS,   AND   COTANGENTS. 


146° 


' 

Sine. 

D.  1'. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

9.736109 
.736303 
.736498 
.736692 

3.23 
3.25  | 
3.23 

9.923591 
.923509 
.923427 
.923345 

1.37 
.37 
.37 

a  7 

9.812517 
.812794 

.813070 
.813347 

4.02 
4.60 
4.62 

10.1S7J83 
.186668 

60 
59 
58 
67 

4 

.736886 

Q  9^ 

.923203 

1 

37 

.813623 

4.60 
4  fin 

.186377 

56 

5 
6 

.737080 
.737274 

3^23 

399 

.9231!  1 
.9230U8 

.818899 

.814176 

.ou 
4.62 

.186101 

55 
54 

7 

.737467 

.*<* 

.92301(5 

1  oft 

.814452 

4.60 

1  185548 

58 

8 
9 

.737661 
.737855 

3.23 
3.23 

.922933 
.922851 

J  .00 
1.37 

1OQ 

.814728 
.815004 

4.60 
4.60 

.184996 

52 
51 

10 

.738048 

3.22 
3.22 

.922768 

.OO 

1.37 

.815280 

4.60 
4.58 

.  1847','U 

50 

11 

9.738241 

399 

9.922686 

9.815555 

4  on 

10.184445 

49 

12 

.738434 

.44 

.922603 

1  .38 

.815831 

.OU 

.184169 

48 

13 
14 

.7:38627 
.738820 

3.22 
3.22 

.922520 
.922438 

1  .38 
1.37 

.816107 
.816382 

4.60 
4.58 
4  fin 

.188808 

.183618 

47 
46 

15 

.739013 

3.22 

.922355 

1  .38 

.816658 

.w 

4RQ 

.183342 

45 

16 

.739206 

3.22 

.922272 

1  .38 

.816933 

.OO 

4  fin 

.188067 

44 

17 

.7'39398 

3.20 

.922189 

i  .00 

.817209 

.DU 

.182791 

4:} 

18 
19 

.739590 
.739783 

3.20 
3.22 

.922106 
.922023 

1  .38 
1.38 

1OQ 

.817484 
.817759 

4.58 
4.58 
4  fin 

.188516 
.188241 

41 

20 

.739975 

3.20 
3.20 

.921940 

.00 

1.38 

.818035 

.ou 
4.58 

.181965 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 

9.740167 
.740359 
.740550 
.740742 
.740934 
.741125 
.741316 
.741508 
.741699 

3.20 
3  18 
3.20 
3.20 
3.18 
3.18 
3.20 
3.18 

9.921857 
.921774 
.921691 
.921607 
.921524 
.921441 
.921357 
.921274 
.921190 

1.38 
1.38 
1.40 
1.38 
1.38 
1.40 
1.38 
1.40 

9.818310 
.818585 
.818860 
.819135 
.819410 
.819684 
.819959 
.820234 
.820508 

4.58 
4.58 
4.58 
4.58 
4.57 
.  4.58 
4.58 
4.57 

10.181690 
.181415 
.181140 
.180865 
.180580 
.180316 
.180041 
.179766 
.179492 

39 
38 

36 
86 
M 
88 

32 
31 

30 

.741889 

3.17 
3.18 

.921107 

1AO 

.820783 

4^57 

.179217 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.742080 
.742271 
.742462 
.742652 
.742842 
.743033 
.743223 
.743413 
.743602 
.743792 

3.18 
3.18 
3.17 
3.17 
3.18 
3.17 
3.17 
3.15 
3.17 
3.17 

9.921023 
.920939 
.920856 
.920772 
.920688 
.920604 
.920520 
.920436 
.920352 
.920268 

1.40 
1.38 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 

9.821057 
.821332 
.821606 
.821880 
.822154 
.822429 
.822703 
.822977 
.823251 
.823524 

4.58 
4.57 
4  57 
4.57 
4.58 
4.57 
4.57 
4.57 
4.55 
4.57 

10.178943 
.178668 
.178:394 
.178120 
.177846 
.177571 
.177^97 
.177023 
.176749 
.176476 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.743982 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 
.745306 
.745494 
.745683 

9.745871 
.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747374 
9.747562 

3.15 
3.17 
3.15 
3  15 
3.15 
3.15 
3.15 
3.13 
3.15 
3.13 

3.15 
3.13 
3  13 
3.13 
3.13 
3.12 
3.13 
3.12 
3,13 

9.920184 
.920099 
.920015 
.919931 
.919846 
.919762 
.919677 
.919593 
.919508 
.919424 

9.919339 
.919254 
.919169 
.919085 
.919000 
.918915 
.918830 
.918745 

9  '.91  8574 

1.42 
1.40 
1.40 
1.42 
1.40 
1.42 
1.40 
1.42 
1.40 
1.42 

1.42 
1.42 
1.40 
1.42 
1.42 
1.42 
1.42 
1.48 
1.42 

9.823798 
.824072 
.824345 
.824619 
.824893 
.825166 
.825439 
.825713 
.825986 

'.  826803 
.827078 
.827351 
827684 

.827897 
.828170 

4.57 
4.55 
4.57 
4.57 
4.55 
4.55 
4.57 
4.55 
4.55 
4.55 

4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.53 

4.53 

10.176202 
.175928 
.175655 
.175381 
.175107 
.174834 
.174561 
.174287 
.171011 
.173741 

10.173468 
.173195 

.172649 
.172103 

10.171013 

19 
18 
17 
16 
15 
14 
13 

n 

n 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

/ 

Cosine. 

D.  1". 

Sine. 

D.  r. 

('otang. 

D.  r. 

Tang. 

r 

66' 

209 


34° 


TABLE   XII. LOGARITHMIC    SINES, 


145° 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.747562 
.7477'49 

3.12 

9.918574 
.918489 

1.42 

9.828987 
.829260 

4.55 

10.171013 
.  170740 

60 

59 

2 
3 

.747936 
.748123 

3.12 
3.12 
3  12 

.918404 
.918318 

1  .42 
1.43 

1  49 

.829532 

.829805 

4.53 
4.55 

4  to 

.170468 
.170195 

58 
57 

4 
5 

6 

8 
9 
10 

.748310 
.748497 
.748683 
.748870 
.749056 
.749243 
.749429 

3.12 
3.10 
3.12 
3.10 
3.12 
3.10 
3.10 

.918233 
.918147 
.918062 
.917976 
.'917891 
.917805 
.917719 

1  .4/6 

1.43 
1.42 
1.43 
1.42 
1.43 
1.43 
1.42 

.830077 
.830349 
.830621 
.830893 
.831165 
.831437 
.831709 

.00 

4.53 
4.53 
4.53 
4.53 
4.53 
4.53 
4.53 

.169923 
.169651 
.169379 
.169107 

.168835 
.168563 
.168291 

56 
55 
54 
53 

52 
51 
50 

11 
12 
13 

9.749615 

.749801 
.749987 

3.10 
3.10 

3AQ 

9.917634 
.917548 
.917462 

1.43 
1.43 

9.831981 
.832253 
.832525 

4.53 
4.53 

10.168019 
.167747 
.167475 

49 

48 
47 

14 
15 

750172 
!  750358 

.Uo 
3.10 
3ns 

.917376 
.917290 

1  .43 
1.43 

.832796 
.833068 

4.52 
4.53 

.167204 
.166932 

46 
45 

16 
17 

.750543 
.  750729 

.Uo 
3.10 
3ns 

.917204 
.917118 

1  .43 
1.43 

.833339 
.833611 

4.52 
4.53 

.166661 
.166389 

44 
43 

18 
19 

.750914 
.751099 

.Uo 
3.08 

3  no 

.917032 
.916946 

1  .43 
1.43 

.833882 
.834154 

4.52 
4.53 

.166118 
.165846 

42 

41 

20 

.751284 

.Uo 

3.08 

.916859 

1  .45 
1.43 

.834425 

4.52 
4.52 

.165575 

40 

21 
22 
23 

24 
25 
26 
27 
28 
29 
30 

9.751469 
.751654 
.751839 
.752023 
.752208 
.752392 
.752576 
.752760 
.752944 
.753128 

3.08 
3.08 
3.07 
3.08 
3.07 
3.07 
3.07 
3.07 
3.07 
3.07 

9.916773 

.916687 
.916600 
.916514 
.916427 
.916341 
.916254 
.916167 
.916081 
.915994 

1.43 
1.45 
1.43 
1.45 
1.43 
1.45 
1.45 
1.43 
1.45 
1.45 

9.834696 
.834967 
.835238 
.835509 
.835780 
.836051 
.836322 
.836593 
.836864 
.837134 

4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.52 
4.50 
4.52 

10.165304 
.165033 
.164762 
.164491 
.164220 
.163949 
.163678 
.163407 
.163136 
.162866 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 

9.753312 
.753495 
.753679 
.753862 
.754046 
.754229 
.754412 

3.05 
3.07 
3.07 
3.07 
3.05 
3.05 

9.915907 
.915820 
.915733 
.915646 
.915559 
.915472 
.915385 

1.45 
1.45 
1.45 
1.45 
1.45 
1.45 

9.837405 
.837675 
.837946 
.838216 

.838487 
.838757 
.839027 

4.50 
4.52 
4.50 
4.52 
4.50 
4.50 

10.162595 
.162325 
.162054 
.161784 
.161513 
.161243 
.160973 

29 
28 
27 
26 
25 
24 
23 

38 
39 
40 

.754595 

.754778 
.754960 

3.05 
3.05 
3.03 
3.05 

.915297 
.915210 
.915123 

1.47 
1.45 
1.45 

1.47 

.839297 
.839568 
.839838 

4.50 
4.52 
4.50 
4.50 

.160703 
.160432 
.160162 

22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.755143 

.755326 
.755508 
.755690 
.755872 
.756054 
.756236 
.756418 
.756600 
.756782 

3.05 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.02 

9.915035 
.914948 
.914860 
.914773 
.914685 
.914598 
.914510 
.914422 
.914334 
.914246 

1.45 
1.47 
1.45 
1.47 
1.45 
1.47 
1.47 
1.47 
1.47 
1.47 

9.840108 
.840378 
.840648 
.1,40917 
.841187 
.841457 
.841727 
.841996 
.842266 
.842535 

4.50 
4.50 
4.48 
4.50 
4.50 
4.50 
4.48 
.4.50 
4.48 
4.50 

10.159892 
.159622 
.159352 
.159083 
.158813 
.158543 
.158273 
.158004 
.157734 
.157465 

19 

18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 

9.756963 
.757144 
.757326 
.757507 

3.02 
3  03 
3.02 
3  no 

9.914158 
.914070 
.913982 
.913894 

1.47 
1.47 
1.47 

9.842805 
.843074 
.843343 
.843612 

4.48 
4.48 
4.48 

10.157195 
156926 
.156657 
.156388 

9 

8 
7 
6 

55 
56 
57 

.757688 
.757869 
.758050 

.U/v 

3.02 
3.02 

.913806 
.913718 
.913630 

1.47 
1.47 
1.47 

.843882 
.844151 
.844420 

4.50 
4.48 
4.48 

.156118 
.155849 
.155580 

5 
4 
3 

58 
59 
60 

.758230 
.758411 
9.758591 

3.00 
3.02 
3.00 

913541 
.913453 
9.913365 

1  .48 
1.47 
1.47 

.844689 
.844958 
9.845227 

4.48 
4.48 
4.48 

.155311 
.155042 
10.154773 

2 
1 
0 

' 

Cosine. 

D.  1".  | 

Sine.  1  D.  1".  i 

Cotang.   D.  1*. 

Tang. 

' 

210 


$5" 


35° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


144° 


' 

Sine. 

D.  1'. 

Cosine. 

D.  r. 

'  Tang. 

D.  1'. 

Ootaag, 

/ 

0 

9.758591 

3  0° 

9.0133(55 

1  48 

9.845287 

44ft 

10.154778 

60 

1 

2 
3 

4 
5 
6 

7 

.758772 
.758952 
.759132 
.759312 
.759492 
.759672 
.759852 

S^OO 
3.00 
3.00 
3.00 
3.00 
3.00 

.913276 
.913187 
.913099 
.913010 
.912922 
.912833 
.912744 

1.48 
1.47 
1.48 
1.47 
1.48 
1.48 

1  4ft 

'.845764 
.846033 
.846302 
.846570 
.846839 
.847108 

4.47 
4.48 
4.48 
4.47 
4.48 
4.48 

.154504 
.154286 

.158696 
.153430 
.158161 

59 

57 
56 
55 
54 

53 

8 
9 

.760031 
.760211 

3.00 

2Qft 

.912655 
.912566 

1.48 

14ft 

.847376 
.847644 

4.47 

44ft 

.152624 
.152856 

52 
51 

10 

.760390 

2.98 

.912477 

1.48 

.847913 

4.47 

.150067 

50 

11 

9.760569 
.760748 

2.98 

9.912388 
.912299 

1.48 

9.848181 
.848449 

4.47 

10.151819 
.151651 

49 

48 

13 
14 
15 

.760927 
.761106 
.761285 

2.98 
2.98 
2.98 

.912210 
.912121 
.912031 

1.48 
1.48 
1.50 

14ft 

.848717 
.848986 
.849254 

4.48 
4.47 

447 

.151283 
.151014 
.150746 

47 
40 
45 

16 
17 
18 
19 
20 

.761464 
.761642 
.761821 
.761999 
.762177 

2.97 
2.98 
2.97 
2.97 
2  98 

.911942 
.911853 
.911763 
.911674 
.911584 

1.48 
1.50 
1.48 
1.50 
1.48 

.849522 
.849790 
.850057 
.850325 
.850593 

4.47 
4.45 
4.47 
4.47 
4.47 

.150478 
.150210 
.149943 
.149675 
.149407 

44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.762356 
.762534 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 
.763954 

2.97 
2.97 
2.95 
2.97 
2.97 
2.95 
2.97 
2.95 
2.95 
2  95 

9.911495 
.911405 
.911315 
.911226 
.911136 
.911046 
.910956 
.910866 
.910776 
.910686 

1.50 
1.50 
1.48 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 

9.850861 
.851129 
.851396 
.851664 
.851931 
.852199 
.852466 
.852733 
.853001 
.853268 

4.47 
4.45 
4.47 
4.45 
4.47 
4.45 
4.45 
4.47 
4.45 
4.45 

10.149139 
.148871 
.148604 
.148336 
.148069 
.147801 
.147534 
.147267 
.146999 
.146732 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.764131 
.764308 
.764485 
.764662 
.764838 
.765015 
.765191 
.765367 
.765544 
.765720 

2.95 
2.95 
2.95 
2.93 
2.95 
2.93 
2.93 
2.95 
2.93 
2  93 

9.910596 
.910506 
.910415 
.910325 
.910235 
.910144 
.910054 
.909963 
.909873 
.909782 

1.50 
1.52 
1.50 
1.50 
1.52 
1.50 
1.52 
1.50 
1.52 
1  52 

9.853535 
.853802 
.854069 
.854336 
.854603 
.854870 
.855137 
.855404 
.855671 
.855938 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.43 

10.146465 
.146198 
.145981 
.145664 
.145397 
.145130 
.144863 
.144596 
.144329 
.144062 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.765896 
.766072 
.766247 
.766423 
.766598 
.766774 
.766949 
.767J24 
.767300 
.767475 

2.93 
2.92 
2.93 
2.92 
2.93 
2.92 
2.92 
2.93 
2.92 
2  90 

9.909691 
.909601 
.909510 
.909419 
.909328 
.909237 
.909146 
.909055 
.908964 
.908873 

1.50 
1.52 
1.52 
1.52 
1.52 
1.52 
1.52 
1.52 
1.52 
1  53 

9.856204 
.856471 
.856737 
.857004 
.857270 
.857537 
.857803 
.858069 
.858336 
.858602 

4.45 
4.43 
4.45 
4.43 
4.45 
4.43 
4.43 
4.45 
4.43 
4.43 

10.143796 

J48263 

.142996 
.142780 
.142463 
.142197 
.141981 
!l41664 
.1413'JS 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.767649 
.767824 
.767999 
.768173 
.768348 
.768522 
.768697 
.768871 
.769045 
9.769219 

2.92 
2.92 
2.90 
2.92 
2.90 
2.92 
2.90 
2.90 
2.90 

9.908781 
.908690 
.908599 
.908507 
.908416 

!  908141 
.908049 

1.52 
1.52 
1.53 
1.52 
1.53 
1.52 
1.53 
1.53 
1.52 

9.858868 
.859134 
.859400 
.859666 
.859932 
960198 
.860464 
.860730 
.860995 
9.861261 

4.43 
4.48 
4  43 
4.43 
4.48 
443 
4.43 
4.42 
4.43 

10.141132 
.140866 

.140600 

.140068 
!  189609 
J80686 

.139270 
.139005 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

125' 


211 


36° 


TABLE   XII. — LOGARITHMIC   SINES, 


143° 


' 

Sine. 

D.I". 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 

4 
5 
6 
7 
8 

9.769219 
.769393 
.769566 
.769740 
.769913 
.770087 
.770260 
.770433 
.770606 

2.90 

2.88 
2.90 
2.88 
2.90 
2.88 
2.88 
2.88 

9.907S58 
.907866 
.907774 
.907682 
.907550 
.907498 
.907406 
.907314 
.907222 

1.53 
1.53 
1.53 
1.53 
1.53 
1.53 
1.53 
1.53 

9.861261 
.861527 
.861792 
.862058 
.862323 
.862589 
.862854 
.863119 
.863385 

4.43 
4.42 
4.43 
4.42 
4.43 
4.42 
4.42 
4.43 

10.138739 
.138473 
.138208 
.137942 
.137677 
.137411 
.137146 
.136881 
.136615 

60 
59 
58 
57 
56 
55 
54 
53 
52 

9 
10 

.770779 
.770952 

2.88 
2.88 
2.88 

.907129 
.907037 

1.55 
1.53 
1.53 

.863650 
.863915 

4.42 
4.42 
4.43 

.136350 
.136085 

51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

9.771125 
.771298 
.771470 
.771643 
.771815 
.771987 
.772159 
.772331 
.772503 

2.88 
2.87 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 

9.906945 
.906852 
.906760 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 

1.55 
1.53 
1.55 
1.53 
1.55 
1.55 
1.55 
1.53 

9.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 

4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 

10.135820 
.135555 
.135290 
.135025 
.134760 
.134495 
.134230 
.133965 
.133700 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

.772675 

2.87 
2.87 

.906111 

1.55 
1.55 

.866564 

4.40 
4.42 

.133436 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.772847 
.773018 
.773190 
.773361 
.773533 
.773704 
.773875 
.774046 
.774217 
.774388 

2.85 

2.87 
2.85 
2.87 
2.85 
2.85 
2.85 
2.85 
2.85 
2.83 

9.906018 
.905925 
.905832 
.905739 
.905645 
.905552 
.905459 
.905366 
.905272 
.905179 

1.55 
1.55 
1.55 
1.57 
1.55 
1.55 
1.55 
1.57 
1.55 
1.57 

9.866829 
.867094 
.867358 
.867623 
.867887 
.868152 
.868416 
.868680 
.868945 
.869209 

4.42 
4.40 
4.42 
4.40 
4.42 
4.40 
4.40 
4  42 
4.40 
4.40 

10.133171 
.132906 
.132642 
.132377 
.132113 
.131848 
.131584 
.131320 
.131055 
.130791 

39 
38 
37 
36 
35 
34 
33 

31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.774558 
.774729 
.774899 
.775070 
.775240 
.775410 
.775580 
.775750 
.775920 
.776090 

2.85 
2.83 
2.85 
2.83 
2.83 
2.83 
2.83 
2.83 
2.83 
2.82 

9.905085 
.904992 
.904898 
.904804 
.904711 
.904617 
.904523 
.904429 
.904335 
.904241 

1.55 
1.57 
1.57 
1.55 
1.57 
1.57 
1.57 
1.57 
1.57 
1.57 

9.869473 
.861)737 
.870001 
.870265 
.870529 
.870793 
.871057 
.871321 
.871585 
.871849 

4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.38 

10.130527 
.130263 
.129999 
.129735 
.129471 
.129207 
.128943 
.128679 
.128415 
.128151 

29 

28 
27 
26 
25 
24 
23 
22 
21 
20 

41 

9.776259 

200 

9.904147 

9.872112 

4  /in 

10.127888 

19 

42 
43 

.776429 
.776598 

.00 
2.82 

2  DO 

.904053 
.903959 

l!57 

1KQ 

.872376 
.872640 

.4U 

4.40 

4QQ 

.127624 
J  27360 

18 
17 

44 
45 

46 

.776768 
.776937 
.777106 

.OO 

2.82 
2.82 

.903864 
.903770 
.903676 

.Do 

1.57 
1.57 

.872903 
.873167 
.873430 

.OO 

4.40 
4.38 

.127097 
.126833 
.126570 

16 
15 
14 

47 
48 

.777275 
.777444 

2^82 

2Qp 

.903581 
.903487 

1  .58 
1.57 
1  58 

.873694 
.873957 

4.40 
4.38 

4QQ 

.126306 
.126043 

13 

12 

49 

50 

.777613 

.777781 

.  O& 

2.80 

2.82 

.903392 
.903298 

1.'57 
1.58 

.874220 

.874484 

.OO 

4.40 
4.38 

.125780 
.125516 

11 
10 

51 

9.777950 

CO 

9.903203 

IRQ 

9.874747 

4OQ 

10.125253 

9 

52 

.778119 

OA 

.903108 

.Do 

.875010 

.OO 

.124990 

8 

53 
54 

.778287 
.778455 

2.80 

2.80 
209 

.9030*4 
.902919 

1.57 
1.58 
1  58 

.875273 

.875537 

4.38 
4.40 

4OQ 

.124727 
.124463 

7 
6 

55 
56 

.778624 

.778792 

.0/5 
2.80 

.902824 
.902729 

l.&B 

.875800 
..876063 

.OO 

4.38 

.124200 
.123937 

5 
4 

57 
58 

.778960 
.779128 

2.80 
2.80 

.902634 
.902539 

l!58 
1  58 

.876326 
.876589 

4.38 
4.38 

4OQ 

.123674 
.123411 

3 
2 

59 

.779295 

Ort 

.902444 

.876852 

.  OO 

.123148 

1 

60 

9.779463 

2.oO 

9.902349 

1.58 

9.877114 

4.37 

10.122886 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang.  1  D.  1'. 

Tang. 

' 

126° 


212 


COSINES,    TANGENTS,    AND   COTANGENTS. 


142° 


' 

Sine. 

D.  1'. 

Cosine. 

D.I'. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 

3 
4 
5 
6 

9.779463 
.779631 
.779798 
.779966 
.780133 
.780300 
.780467 

2.80 
2.78 
2.80 
2.78 
2.78 
2.78 

9.902349 
.902253 
.902158 
.902063 
.901967 
.901872 
.901776 

1.60 
1.58 
1.58 
1.60 
1.58 
1.60 

9.877111 

.877640 
.877903 
.878165 
.878428 
.878691 

4.38 
4.38 
4.38 
4.37 

4.38 

4.  as 

10.122886 

.128087 

.121835 
.121578 

.121309 

60 
59 

58 

56 
66 

54 

7 
8 
9 

10 

.780634 
.780801 
.780968 
.781134 

2.78 
2.78 
2.77 
2.78 

.901681 
.901585 
.901490 
.901394 

1.60 
1.58 
1.60 
1.60 

.878953 
.879216 
.879478 
.879741 

4.38 
4.37 
4.38 
4.37 

.121047 
.120784 
.120522 

.120259 

68 

68 
61 

50 

11 
12 
13 

9.781301 
.781468 
.781634 

2.78 

2.77 

277 

9.901298 
.901202 
.901106 

1.60 
1.60 

9.880003 
.880265 
.880528 

4.37 
4.38 

10.119997 
.110786 

.119472 

49 
48 
47 

14 
15 
16 
17 
18 
19 

.781800 
.781966 
.782132 
.782298 
.782464 
.782630 

2.77  ! 
2.77  ! 
2.77  i 
9  77 

2>7 

2C-7 

.901010 
.900914 
.900818 
.900722 
.900626 
.900529 

1.60 
1.60 
1.60 
1.60 
1.62 

1  Art 

.880790 
.881052 
.881314 
.881577 
.881839 
.882101 

4.37 
4.37 
4.38 
4.37 

4.37 
407 

.119210 
.118948 
,118686 
.118423 
.118161 
.117899 

46 
45 
44 
43 
42 
41 

20 

.782796 

2.75 

.900433 

1.60 

.882363 

4.37 

.117637 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 

9.782961 
.783127 

.783292 
.783458 
.783623 
.783788 
.783953 
.784118 
.784282 

2.77 
2.75 
2.77 
2.75 
2.75 
2.75 
2.75 
2.73 

9.900337 
.900240 
.900144 
.900047 
.899951 
.899854 
.899757 
.899660 
.899564 

1.62 
1.60 
1.62 
1.60 
1.62 
1.62 
1.62 
.60 

9.882625 

.882887 
.883148 
.883410 
.883672 
.883934 
.884196 
.884457 
.884719 

4.37 
4.35 
4.37 
4.37 
4.37 
4.37 
4.35 
4.37 

4  OK 

10.117375 
.117113 
.116852 
.116590 
.116328 
.116066 
.115804 
.115543 
.115281 

39 
38 
37 
36 
35 
34 

as 

32 
31 

30 

.784447 

2.75 
2.75 

.899467 

.62 

.884980 

4.37 

.115020 

30 

31 
32 
33 
34 
35 

9.784612 
.784776 
.784941 
.785105 
.785269 

2.73 
2.75 
2.73 
2.73 

9.899370 
.899273 
.899176 
.899078 
.898981 

.62 
.62 
.63 
.62 

9.885242 
.885504 
.885765 
.886026 
.886288 

4.37 
4.35 
4.35 
4.37 

4  OK 

10.114758 
.114496 
.114235 
.113974 
.113712 

29 
28 
27 
26 
25 

36 
37 
38 
39 
40 

.785433 

.785597 
.785761 
.785925 
.786089 

2.73 
2.73 
2.73 
2.73 
2.73 
2.72 

.898884 
.898787 
.898689 
.89&592 
.898494 

.62 
.63 
.62 
.63 
.62 

.886549 
.886811 
.887072 
.887333 
.887594 

4.37 
4.35 
4.35 
4.35 
4.35 

.113451 
.113189 
.112928 
.112667 
.112406 

24 
23 
22 
21 
20 

41 
42 
43 

44 
45 
46 

47 
48 
49 
50 

9.786252 
.786416 
.786579 
.786742 
.786906 
.787069 
.787232 
.787395 
.787557 
.787720 

2.73 
2.72 
2.72 
2.73 
2.72 
2.72 
2.72 
2.70 
2.72 
*  72 

9.898397 
.898299 
.898202 
.898104 
.898006 
.897908 
.897810 
.897712 
.897614 
.897516 

.63 
.62 
.63 
.63 
.63 
.63 
.63 
.63 
.63 
63 

9.887855 
.888116 
.888378 
.888639 
.888900 
.889161 
.889421 
.889682 
.889943 
.890204 

4.35 
4.37 
4.35 
4.35 
4.35 

4.  as 

4.35 
4.35 
4.35 
4.35 

10.112145 
.111884 
.111622 
.111361 
.111100 
.110839 
.110579 
.110318 
.110057 
.109796 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.787883 
.788045 
.788208 
.788370 
.788532 
.788694 
.788856 
.789018 
.789180 
9.789342 

2.70 
2.72 
2.70 
2.70 
2.70 
2  70 
2.70 
2.70 
2.70 

9.897418 
.897320 
.897222 
.897123 
.897025 
.896926 
.896828 
.896729 
.896631 
9.896532 

.63 
.63 
.65 
.63 
.65 
.63 
1.65 
1.63 
1.65 

9.890465 
.890725 

.89098(5 
.891247 
.891507 
.891768 

!  892289 
.892549 
9.892810 

1 

4.33 
4.35 
4.35 

4.  as 

4.35 

4.  as 
4.  as 

10.109535 
.109275 
.109014 
1108758 
.106188 
.108888 
!  107878 
.107711 
.107451 
10.107190 

9 

8 

7 
6 
5 
4 
3 
2 
1 
0 

"/" 

Cosine. 

D  1". 

Sine. 

D.  r. 

i  Cotang. 

D.  r. 

Tang. 

' 

127° 


213 


52' 


38° 


TABLE  XII. — LOGARITHMIC    SINES, 


141° 


' 

Sine. 

D.  1*. 

Cosine. 

D.  1". 

Tang. 

D.I". 

Cotang. 

• 

0 

1 

2 
3 
4 
5 

9.789342 
.789504 
.789665 
.789827 
.789988 
.790149 

2.70 

2.68 
2.70 
2.68 
2.68 

9.896532 
.896433 
.896335 
.896236 
.896137 
.896038 

1.65 
1.63 
1.65 
1.65 
1.65 

CK 

9.892810 
.893070 
.893331 
.893591 
.893851 
.894111 

4.33 
4.35 
4.33 
4.33 
4.33 

4  OR 

10.107190 
.106930 
.106669 
.106409 
.106149 
.105889 

GO 
59 
58 
57 
56 
55 

6 

.790310 
.790471 

2^68 

2  CO 

.895939 
.895840 

.  .DO   , 

.894372 
.894632 

.00 

4.33 

4qo 

.105628 
.105368 

54 
53 

8 

.790632 

.  Do 

2  AS 

.895741 

67 

.894892 

.  o<* 
4qq 

.105108 

52 

9 
10 

.790793 
.790954 

.Do 

2.68 
2.68 

.895641 
.895542 

f 

.65 
.65 

.895152 
.895412 

.00 

4.33 
4.33 

.104848 
.104588 

51 
50 

11 

9.791115 

9.895443 

R7 

9.895672 

400 

10.104328 

49 

12 
13 
14 
15 
16 
17 
18 
19 

.791275 
.791436 
.791596 
.791757 
.791917 
.792077 
.792237 
.792397 

2^68 
2.67 
2.68 
2.67 
2.67 
2.67 
2.67 

2A7 

.895343 
.895244 
.895145 
.895045 
.894945 
.894846 
.894746 
.894646 

.Of 

.65 
.65 
.67 
.67  ! 
.65 
.67 

:  .67 

.895932 
.896192 
.896452 
.896712 
.896971 
.897231 
.897491 
.897751 

.00 

4.33 
4.33 
4.33 
4.32 
4.33 
4.33 
4.33 

.104068 
.103808 
.103548 
.103288 
.103029 
.102769 
.102509 
.102249 

48 
47 
46 
45 
44 
43 
42 
41 

20 

.792557 

.  Of 

2.65 

.894546 

:!e? 

.898010 

4^33 

.101990 

40 

21 

22 
23 
24 
25 
26 
27 

9.792716 
.792876 
.793035 
.793195 
.793354 
.793514 
.793673 

2.67 
2.65 
2.67 
2.65 
2.67 
2.65 

9.894446 
.894346 
.894246 
.894146 
.894046 
.893946 
.893846 

1.67 
,  1.67 
1.67 
1.67 
1.67 
1.67 

9.898270 
.898530 
.898789 
.899049 
.899308 
.399568 
.899827 

4.33 
4.32 
4.33 
4.32 
4.33 
4.32 

10.101730 
.10147C 
.101211 
.100951 
.100692 
.100432 
.100173 

39 
38 
37 
36 
35 
34 
33 

28 

.793832 

2.65 

.893745 

1  .68 

.900087 

4.33 

.099913 

32 

29 

.793991 

2.65 

2   Of 

.893645 

.67 

.900346 

4.32 
409 

.099654 

31 

30 

.794150 

.DO 
2.63 

.893544 

!67 

.900605 

,(S6 

4.32 

.099395 

30 

31 

9.794308 

9.893444 

9.900864 

400 

10.099136 

29 

32 
33 

.794467 
.794626 

2.65 
2.65 

.893343 
.893243 

'.67 

AQ 

.901124 
.901383 

.00 

4.32 

.098876 
.098617 

23 
27 

34 

.794784 

2.63 

.893142 

.Do 

.901642 

^'e\n 

.098358 

26 

35 
36 

.794942 
.795101 

2.63 
2.65 

.893041 
.892940 

.68 

.68 

.901901 
.902160 

4.O4 

4.32 

.098099 
.097840 

25 
24 

37 
38 

.795259 
.795417 

2.63 
2.63 

.892839 
.892739 

.68 
.67 

.902420 
.902679 

4.33 

4.32 

.097580 
.097321 

23 
22 

39 

40 

.795575 
.795733 

2.63 
2.63 
2.63 

.892638 
.892536 

.68 

.70 
.68 

.902938 
.903197 

4.32 
4.32 

4.32 

.097062 
.096803 

21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 

9.795891 
.796049 
.796206 
.796364 
.796521 
.796679 
.796836 
.796993 
.797150 

2.63 

2.62 
2.63 
2.62 
2.63 
2.62 
2.62 
2.62 

9.8924a5 
.892334 
.892233 
.892132 
.892030 
.891929 
.891827 
.891726 
.891624 

.68 
.68 
.68 
.70 
.68 
.70 
.68 
.70 

Aft 

9.903456 
.903714 
.903973 
.904232 
.904491 
.904750 
.905008 
.905267 
.905526 

4.30 
4.32 
4.32 
4.32 
4.32 
4.30 
4.32 
4.32 

4OO 

10.096544 
.096286 
.096027 
.095768 
.095509 
.095250 
.094992 
.094733 
.094474 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.797307 

2!  62 

.891523 

.Do 

.70 

.905785 

.Die 

4.30 

.094215 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.797464 
.797621 

.797777 
.797934 
.798091 
.798247 
.798403 
.798560 
.798716 
9.79887'2 

2.62 

2.60 
2.62 
2.62 
2.60 
2.60 
2.62 
2.60 
2.60 

9.891421 
.891319 
.891217 
.891115 
.891013 
.890911 
.890809 
.890707 
.890605 
9.890503 

.70 
.70 
.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70  , 

9.906043 
.906302 
.906500 
.906819 
|  .907077 
.907336 
.907594 
.907853 
.908111 
9.908369 

4.32 
4.30 
4.32 
4.30 
4.32 
4.30 
4.32 
4.30 
4.30 

10.093957 
.093698 
.093440 
.093181 
.092923 
.092664 
.092406 
.092147 
.091889 
10.091631 

9 

8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  i".  i 

Cotang. 

D.  r. 

Tang. 

' 

128° 


214 


COSINES,    TANGENTS,    AND    COTANGENTS. 


140° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

/ 

0 

9.798872 

2  60 

9.890503 

1  72 

9.908369 

4  o0   10.MH1,:',! 

60 

1 

.799028 

2  60 

.890400 

1  7fl 

.906628 

rS  i  .«)'.»  i:;:-' 

59 

2 

.799184 

2  58 

.890298 

1  .  I\J 

!X>KHSI)   1'™  !   O'jnit 

58 

3 

4 
5 

.799339 
.799495 
.799651 

2^60 
2.60 
2  58 

.890195 
.890093 
.889990 

l'.70 
1.72 
1  70 

'.909144 

.909402 
.909660 

4.30 
4.30 

4OA 

.1  '.MS.M1 

.090696 
.090840 

57 
56 
56 

6 

.799806 

.889888 

I'M 

.909918 

.oU 

.090068 

54 

r» 

.799962 

2.60 

.889785 

l!72 

.910177 

4.32 

4OA 

58 

8 

.800117 

jj'jjS 

.889682 

.910435 

.ou 

58 

9 
10 

.800272 
.800427 

2  '.58 
2.58 

.889579 
.889477 

l'.70 
1.72 

.910693 
.910951 

4.30 
4.30 
4.30 

.069807 

51 

50 

11 
12 
13 
14 

9.800582 
.800737 
.800892 
.801047 

2.58 
2.58 
2.58 

9.889374 

.889271 
.889168 
.889064 

1.72 
1.72 
1.73 

1  72 

9.911209 
.911467 
.911725 
.911982 

4.30 
4.30 
4.28 
4  30 

10.088791 
066583 

49 
48 
47 
46 

15 
16 
17 
18 
19 
20 

.801201 
.801356 
.801511 
.801665 
.801819 
.801973 

2  '.58 
2.58 
2.57 
2.57 
2.57 
2.58 

.888961 
.888858 
.888755 
.888651 
.888548 
.888444 

1>2 
1.72 
1.73 
1.72 
1.73 
1.72 

.912240 
.912498 
.912756 
.913014 
.913271 
.913529 

4.'  30 
4.30 
4.30 
4.28 
4.30 
4.30 

.067760 
.067508 

!  086988 

.086471 

4B 
44 
48 

41 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.802128 
.802282 
.802436 
.802589 
.802743 
.802897 
.803050 
.803204 
.803357 
.803511 

2.57 
2.57 
2.55 
2.57 
2.57 
2.55 
2.57 
2.55 
2.T.7 
2.55 

9.888341 
.888237 
.888134 
.888030 
.887926 
.887822 
.887718 
.887614 
.887510 
.887406 

1.73 
1.72 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 

9.913787 
.914044 
.914302 
.914560 
.914817 
.915075 
.915332 
.915590 
.915847 
.916104 

4.28 
4.30 
4.30 
4.28 
4.30 
4.28 
4.30 
4.28 
4.28 
4.30 

10.086213 
065956 
.085698 
.065440 

.OK51S3 
.084925 
.084668 
.084410 
.084153 
.083896 

39 
38 
37 
36 
35 
84 
33 
32 
31 
30 

31 
32 
33 
34 

9.803664 
.803817 
.803970 
.804123 

2.55 
2.55 
2.55 
2  55 

9.887302 
.887198 
.887093 
.886989 

1.73 
1.75 
1.73 
1.73 

9.916362 
.916619 
.916877 
.917134 

4.28 
4.30 
4.28 
4  28 

10.083638 
.083:581 
.083123 
.068866 

29 

28 
27 
86 

35 
36 

.804276 
804428 

2^55 

.886885 
886780 

.917391 
.917648 

4.28 

.IK.WH) 

I068858 

M 

37 
38 
39 
40 

'.804581 
.804734 
.804886 
.805039 

2.55 
2.55 
2.53 
2.55 
2.53 

!  886676 
.886571 
.886466 
.886362 

1.73 
1.75 
1.75 
1.73 
1.75 

.917906 
.918163 
.918420 
-  .918677 

4.30 
4.28 
4.28 
4.28 
4.28 

.068094 

.081837 
.081580 
.081323 

23 

a 

20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.805191 
.805343 
.805495 
.805647 
.805799 
.805951 
.806103 
.806254 
.806406 
.806557 

2.53 
2.53 
2.53 
2.53 
2.53 
2.53 
2.52 
2.53 
2.52 
2  53 

9.886257 
.886152 
.886047 
.885942 
.885837 
.885732 
.885627 
.885522 
.885416 
.885311 

1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.77 
1.75 
1.77 

9.918934 
.919191 
.919448 
.919705 
.919962 
.920219 
.920476 
.920733 
.920990 
.921247 

4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.27 

10.081066 
080809 

.060895 

.nsou-is 
.079781 

.079524 

.079010 
.078753 

19 

18 
17 
16 
15 
14 
18 
18 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.806709 
.806860 
.807011 
.807163 
.807314 
.807465 
.807615 
.807766 
.807917 
9.808067 

2.52 
2.52 
2.53 
2.52 
2.52 
2.50 
2.52 
2.52 
2.50 

9.885205 
.885100 
.884994 
.884889 
.884783 
.884677 
.884572 
.884466 
.884360 
9.884254 

1.75 
1.77 
1.75 
1  1.77 
1.77 
1.75 
1.77 
j  1.77 
1.77 

9.921503 
.921760 
.922017 
.988874 
.988580 
.^8787 
.923044 
.923300 

4.28 
4.28 
4.28 
4.«? 

4.28 

4.88 
4.28 

10.07R497 

.077813 

.070700 
.079448 

10.076186 

9 

s 
7 
6 
5 
4 
3 
2 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

129° 


215 


40° 


TABLE   XII. — LOGARITHMIC    SINES, 


139° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 

9.808067 
.808218 
.808368 

2.52 
2.50 
2  52 

9.884254 
.884148 
.884042 

1.77 
1.77 

1  77 

9.923814 
.924070 
.924327 

4.27 
4.28 
4  27 

10.076186 
.075930) 
.075673 

60 

59 
58 

3 

.808519 

.883936 

1.14 

78 

.924583 

.075417 

57 

4 

5 

.808669 
.808819 

2^50 

.883829 
.883723 

.  <O 

'I7 

.924840 
.925096 

4.27 

.075160 
.074904 

56 

55 

6 
7 
8 
9 
10 

.808960 
.809119 
.809269 
.809419 
.809569 

2.50 
2.50 
2.50 
2.50 
2.50 
2.48 

.883617 
.883510 
.883404 
.883297 
.883191 

.77 
.78 
.77 
.78 
.77 
.78 

.925352 
.925609 
.925865 
.926122 
.926378 

4.27 

4.28 
4.27 
4.28 

4.27 
4.27 

.074648 
.074391 
.074135 
.073878 
.073622 

54 
53 
52 
51 
50 

11 
12 

9.809718 
.809868 

2.50 

9.883084 

.882977 

1.78 

9.926634 

.926890 

4.27 

10.073366 
.073110 

49 

48 

13 

.810017 

2.48 

.882871 

1  .77 

.927147 

S 

.072853 

47 

14 
15 
16 
17 
18 
19 
20 

.810167 
.810316 
.810465 
.810614 
.810763 
.810912, 
.811061 

2^48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 

.882764 
.882657 
.882550 
.882443 
.882336 
.882229 
.882121 

1  .78 
1.78 
1.78 
1.78 
.78 
.78 
.80 
.78 

.927403 
.927659 
.927915 
.928171 

.928427 
.928684 
.928940 

4.2< 
4.27 
4.27 

4.27 
4.27 
4.28 

4.27 
4.27 

.072597 
.072341 
.072085 
.071829 
.071573 
.071316 
.071060 

46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 

9.811210 
.811358 
.811507 
.811655 

2.47 
2.48 
2.47 

9.882014 
.881907 
.881799 
.881692 

.78 
.80 
.78 

9.929196 
.929452 
.929708 
.929964 

4.27 
4.27 
4.27 

10.070804 
.070548 
.070292 
.070036 

39 

38 
37 
36 

25 

26 

.811804 
.811952 

2.48 
2.47 

9  47 

.881584 

.881477 

.80 
.78 

•  pn 

.930220 
.930475 

4.27 
4'2£ 

.069780 
.069525 

35 
34 

27 

.812100 

/c.4i 

2  47 

.881369 

.OU 

PA 

.930731 

97 

.069269 

33 

28 

.812248 

.881261 

.OU 

.930987 

.XI 

.069013 

32 

29 
30 

.812396 
.812544 

2.47 
2.47 
2.47 

.881153 
.881046 

.80 
.78 
.80 

.931243 
.931499 

.27 
.27 
.27 

.068757 
.068501 

31 
30 

31 

9.812692 

247 

9.880938 

DA 

9.931755 

10.068245 

29 

32 

.812840 

.44 

2  47 

.880830 

.OU 
PA 

.932010 

27 

.067990 

28 

33 
34 
35 

.812988 
.813135 
.813283 

2^45 

2.47 

.880722 
.880613 
.880505 

.OU 

.82 
.80 
on 

.932266 
.932522 
.932778 

4^27 
4.27 

.067734 
.067478 
.067222 

27 
26 
25 

36 
37 

.813430 
.813578 

2.45 
2.47 

24K. 

.880397 
.880289 

.ou 
.80 

.933033 
.933289 

4.25 

4.27 

497 

.066967 
.066711 

24 

23 

38 

.813725 

.40 

2   AC 

.880180 

PA 

.933545 

,Xt 

4  OK 

.066455 

22 

39 
40 

.813872 
.814019 

.40 

2.45 
2.45 

.880072 
.879963 

.OU 

.82 
.80 

.933800 
.934056 

:.O9 

4.27 
4.25 

.066200 
.065944 

21 
20 

41 
42 
43 

9.814166 
.814313 
.814460 

2.45 
2.45 

9.879855 
.879746 
.879637 

.82 

.82 

PA 

9.934311 
.934567 
.934822 

4.27 
4.25 

497 

10.065689 
.065433 
.065178 

19 

18 
17 

44 

.814607 

?••  j5 

.879529 

.oU 

.935078 

.«< 

.064922 

16 

45 

.814753 

2.4o 

.879420 

.82 

89 

.935333 

4.25 

.064667 

15 

46 

47 

.814900 
.815046 

2^43 

.879311 
.879202 

.0,* 

.82 

89 

.935589 
.935844 

4^25 

497 

.064411 
.064156 

14 
13 

48 
49 

.815193 
.815339 

2^43 

.879093 

.878984 

.0.* 

.82 

.936100 
.936355 

•./ft 

4.25 

.063900 
.063645 

12 
11 

50 

.815485 

2.43 
2.45 

.878875 

.82 

.82 

.936611 

4.27 
4.25 

.063389 

10 

51 
52 
53 
54 

9.815632 

.815778 
.815924 
.816069 

2.43 
2.43 
2.42 

9.878766 
.878656 
.878547 
.878438 

.as 

.82 
.82 

9.936866 
.937121 
.937377 
.937632 

4.25 
4.27 
4.25 

10.063134 
.062879 
.062623 
.062368 

9 

8 
7 
6 

55 
56 

.816215 
.816361 

2.43 
2.43 

.878328 
.878219 

.83 

.82 

.937887 
.938142 

4.25 
4.25 

.062113 
.061858 

5 
4 

57 

.816507 

2.43 

.878109 

.83 

.938398 

4.27 

.061602 

3 

58 
59 

.816652 
.816798 

2.42 
2.43 

.877999 

.877890 

.83 

.82 

.938653 
.938908 

4.25 
4.25 

.061347 
.061092 

2 
1 

60 

9.816943 

2.42 

9.877780 

1  .83 

9.939163 

4.25 

10.060837 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

130° 


216 


49' 


41° 


COSINES,    TANGENTS,   AND    COTANGENTS. 


138° 


,. 

i       | 

' 

Sine. 

D.  1'. 

Cosine. 

D.  r.  ' 

Tang. 

D.  r. 

/ 

1 

o 

9.816943 

o  to   9.877780 

0.030103 

1 

.817088 

;  .8776:0  i  {•'':;   .'.MM  is 

4M-. 

2 

[817233 

,)  |  }•*•* 

3 

.817379 

.877450         .o:500-JS 

4  OR 

4 

.817524 

040   !    •S77"1°    l"S 

5 

.817008 

242    .877230   ,22    .!M»M:i'.) 

4  or 

6 
7 
8 
9 
10 

.817813 
.817953 
.818103 
.818247 
.81839:2 

2'.42 
2.43 

2.40 
2.42 

2.40 

.877120 
.877010 
.870800 
.876789 
.876078 

I  >:{ 

1.85 

1.83 
1.85 
1.83 

.940694 

.010010 

.941804 
.941459 

.941713 

4.25 

4.86 
4.88 
4.85 

.050: 

.066641 

.058287 

64 

68 
51 

50 

11 
12 

9.818536 

818081 

2.42 

9.8705G8 
.870457 

1.85 

9.941968 
.942223 

4.25 

10.058032 

49 

13 
14 
15 
10 

17 
18 
19 
20 

.818825 
.818909 
.819113 
.819257 
.819401 
.819545 
.819689 
.819832 

2.40 
2.40 
2.40 

2.40 
2.40 
2.40 
2.40 
2.38 
2  40  i 

.876347 
.876236 
.876125 
.876014 
.875904 
.875793 
.875682 
.875571 

1.85 

1.85 
1.85 
1.83  . 
1.85 
1.85 
1.85 
1.87 

.942478 
.942788 

.942988 
.943243 
.943498 
.943752 
.944007 
.94-4202 

4.26 
4.86 

4.25 
4.25 
4.23 
4.25 
4.25 
4  25 

.057012 
.056757 
.066608 
.066248 

.055993 
.055738 

47 
46 
45 
41 
43 
42 
41 
40 

21 
22 
23 

24 
25 

26 
27 
28 
29 
30 

9.819976 
.820120 
.820263 
.820406 
.820550 
.820693 
.820836 
.820979 
.821122 
.821265 

2.40 
2.38 
2.38 
2.40 
2.38 
2.38 
2.38 
2.38 
2.38 
2  37 

9.875459 

.875348 
.875237 
.875126 
.875014 
.874903 
.874791 
.874680 
.874568 
.874456 

1.85 
1.85 
1.85 

1.87 
1.85 

1.87 
1.85 
1.87 
1.87 
1  87 

9.944517 
.944771 
.945026 
.945281 
.945535 
'  .945790 
.946045 
.946299 
.946554 
!  .946808 

4.23 
4.25 
4.25 
4.23 
4.25 
4.25 
4.23 
4.25 
4.23 
4.25 

10.055483 

.054974 
.054719 
.054465 
.054210 
.053955 
.053701 
.068446 
.053192 

39 

as 

87 
86 

35 
84 

32 
81 

SO 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.821407 
821550 
.821093 
.821835 
.821977 
.822120 
.822202 
.822404 
.822546 
.822088 

2.38 
2.38 
2.37 
2.37 
2.38 
2.37 
2.37 
2.37 
2.37 
9  37 

9.874341 
.874232 
.874121 
.874009 
.873896 
.873784 
.873672 
.873560 
.873448 
.873335 

1.87 
1.85 
1.87 
1.88 
1.87 
1.87 
1.87 
1.87 
1.88 
1  87 

9.947063 
.947318 
.947572 
.947827 
1  .948081 
.948885 
.948590 
.948844 
.949099 
.949353 

4.25 
4.23 
4.25 
4.23 
4.23 
4.25 
4.23 
4.25 
4.23 
4.25 

10.052937 
.052682 
.052428 
.052173 
.051919 
.051665 
.051410 
.051156 
.060901 
.050647 

29 
28 
27 

25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 

52 

9.822830 
.822972 
.823114 
.823255 
.823397 
.823539 
.823080 
.823821 
.8-2:5003 
.824104 

9.824245 

.824380 

2.37 
2.37 
2.35 
2.37 
2.37 
2.35 
2.35 
2.37 
2.35 
2.35 

2.35 

!  9.873223 
.873110 
1  .872998 
.872885 
.872772 
.872659 
.872547 
.872434 
.872321 
.872208 

9.872095 
I   .871981 

1  88 
1.87 

1.88 
1.88 
1.88 
1.87 
1.88 
1.88 
1.88 
1.88 

1.90 
1  88 

9.949608 
.949862 
.950116 
.950371 
.950625 
.950879 
.951133 
.951388 
.951642 
.951896 

9.952150 
.952405 

4.23 
4.23 
4.25 
4.23 
4.23 
4.23 
4.25 
4.23 
4.23 
4.23 

4.23 

10.050392 
.060188 
[048884 
.048689 

.048181 
.048867 
.048618 

10.047850 
.047595 
047341 

In 

18 
17 
16 
15 
14 
18 
18 
11 
10 

9 

8 

7 

53 
54 
55 
56 
57 
58 
59 
60 

.824527 

.824668 
.834808 
.824949 
.825090 
.825230 
.825371 
9.825511 

2.35 
2.33 
2.35 
2.35 
2.33 
2.35 
2.33 

!  .871868 
.871755 
.871641 
.871528 
.871414 
.871301 
.871187 
9.871073 

1.88 
1.90 
1.88 
1.90 

i.as 

1.90 
1.90 

.952913 
.953167 
;  958481 
.868878 
.968981 
.954183 
9.954487 

4.23 
4.23 
4.23 

4.23 
4.23 

.047087 
.046888 
.046579 
I046886 

1  045817 

6 

5 

4 
8 
2 

1 
0 

1  Cosine. 

D.  1'. 

Sine. 

D.  1".  I  Cotnntf 

D.  r 

Tang.  1  ' 

4R« 

131° 


217 


42° 


TABLE   XII. — LOGARITHMIC    SINES, 


i 

' 

Sine. 

D.  1'. 

Cosine. 

I).  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.825511 

9.871073 

1  ftQ 

9.954437 

4  90 

10.045563 

60 

1 

2 
3 

.825651 
.825791 
.825931 

2!  33 
2.33 

:  .870960 
.870846 
.870732 

1  .00 

1.90 
1.90 

.954691 
.954946 
.955200 

.In) 

4.25 

4.23 

.045309 
.045054 
.044800 

59 
58 
57 

4 
5 
6 

.826071 
.826211 
.826&51 

2.33 
2.33 
2.33 

200 

.870618 
.870504 
.870390 

1.90 
1.90 
1.90 
1  on 

.955454 
.955708 
.955961 

4.23 
4.23 
4.22 
490 

.044546 
.044292 
.044039 

56 
55 
54 

8 

.  9 

10 

.826491 
.826631 

.826770 
.826910 

.  OO 

2.33 
2.32 
2.33 
2.32 

.870276 
.870161 
.870047 
.869933 

i  .yu 
1.92 
1.90 
1.90 
1.92 

.956215 
.956469 
.956723 
.956977 

./OO 

4.23 
4.23 
4.23 
4.23 

.043785 
.043531 
.043277 
.043023 

53 

52 
51 
50 

11 
12 
13 
14 
15 
16 

9.827049 

.827189 
.827328 
.827467 
.827606 
.827745 

2.33 

2.32 
2.32 
2.32 
2.32 

9.869818 
.869704 
.869589 
.869474 
.869360 
.869245 

1.90 
1.92 
1.93 

1.90 
92 

9.957231 
.957485 
.957739 
.957993 

.958247 
.958500 

4.23 
4.23 
4.23 
4.23 

4.22 

10.042769 
.042515 
.042261 
.042007 
.041753 
.041500 

49 
48 
47 
46 
45 
44 

17 

18 
19 

20 

.827884 
.828023 
.828162 
.828301 

2.32 
2.32 
2.32 
2.32 
2.30 

.869130 
.869015 
.868900 
.868785 

.92 
.92 
.92 
.92 
1.92 

.958754 
.959008 
.959262 
.959516 

4.23 
4.23 
4.23 
4.23 
4.22 

.041246 
.040992 
.0407.8 
.040484 

43 

42 
41 
40 

21 

9.828439 

9.868670 

9.959769 

4cyf> 

10.040231 

39 

22 
23 
24 
25 
26 
27 
28 
29 
30 

.828578 
.828716 
.828855 
.828993 
.829131 
.829269 
.829407 
.829545 
.829683 

2!  30 
2.32 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 

.868555 
.868440 
.868321 
.868209 
.868093 
.867978 
.867862 
.867747 
.867631 

1.'92 
1.93 
1.92 
1.93 
1.92 
1.93 
1.92 
1.93 
1.93 

.960023 
.960277 
.960530 
.960784 
.961038 
.961292 
.961545 
.961799 
.962052 

./OO 

4.23 

4.22 
4.23 
4.23 
4.23 
4.22 
4.23 
4.22 
4.23 

.039977 
.039723 
.039470 
.039216 
.038962 
.038708 
.038455 
.038201 
.037948 

38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
88 

0.829821 
.829959 

2.30 

9.867515 
.867399 

1.93 

9.962306 
.962560 

4.23 

10.037694 
.037440 

29 

28 

33 
34 
35 

.830097 
.830234 
.830372 

2  '.28 
2.30 

2Oft 

.867283 
.867167 
.867051 

1.93 
.93 
.93 

no 

.962813 
.963067 
.963320 

4.22 
4.23 

4.22 

.037187 
.036933 
.036680 

27 
26 
25 

36 

.830509 

.«o 

.866935 

.Jo 

no 

.963574 

49°. 

.036426 

24 

37 

.830646 

6.40 

.866819 

.yo 

.963828 

.JSO 

.036172 

23 

38 
39 

.830784 
.830921 

2.30 

2.28 

.866703 
.866586 

.93 
.95 

.964081 
.964335 

4.22 
4.23 

.035919 
.035665 

22 

21 

40 

.831058 

2.28 
2.28 

.866470 

.93 
.95 

.964588 

4^23 

.035412 

20 

41 
42 
43 
44 
45 

9.831195 
.831332 
.831469 
.831606 
.831742 

2.28 
2.28 
2.28 
2.27 

9.866353 
.866237 
.866120 
.866004 

.865887 

.93 
.95 
.93 
.95 

(\K 

9.964842 
.965095 
.965349 
.965602 
.965855 

4.22 
4.23 
4.22 
4.22 
490 

10.035158 
.034905 
.034651 
.034398 
.034145 

19 
18 
17 
16 
15 

46 
47 
48 
49 
50 

.831879 
.832015 
.832152 
.832288 
.832425 

2.  '27 
2.28 
2.27 
2.28 
2.27 

.865770 
.865653 
.865536 
.865419 
.865302 

.yo 
.95 
.95 
.95 
.95 
.95 

.966109 
.966362 
.966616 
.966869 
.967123 

./OO 

4.22 
4.23 
4.22 
4.23 
4.22 

.033891 
.033638 
.033384 
.033131 
.032877 

14 
13 
12 
11 
10 

51 

9-832561 

9  97 

9.865185 

Qt 

9.96737'6 

99 

10.032624 

9 

52 
53 
54 
55 

56 
57 
58 
59 

.832697 
.832833 
..832969 
.833105 
.833241 
.833377 
.833512 
.833648 

Z.-KY 

2.27 
2.27 
2  27 
2^27 
2.27 
2.25 
2.27 

.865068 
.864950 
.864833 
.864716 
.864598 
.864481 
.864363 
.864245 

.yo 
.97 
.95 
.95 
.97 
.95 
.97  i 

:997 

.967629 
.967883 
.968136 
.968389 
!  .968643 
.968896 
.969149 
.969403 

4.  .23 
4.22 
4.22 
4.23 

4.22 
4.22 
4.23 

.032371 
.032117 
.031864 
.031611 
.031357 
.031104 
.030851 
.030597 

8 
7 
6 
5 
4 
3 

1 

60 

9.833783 

2.25 

9.864127 

1  9.969656 

4.22 

10.030344 

0 

1 

Cosine. 

D.I". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

132' 


47- 


43° 


COSINES,    TANGENTS,    AND    COTANGENTS. 


136° 


' 

Sine. 

I),  r. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.833783 

o  o~ 

9.864127 

1  95 

9.969056 

10.030344 

60 

1 

.833919 

2  25  ' 

.804010 

.969909 

.080091 

59 

o 

.834054 

2  ^5 

.8C.3S-92  ' 

,')! 

.970162 

,  T 

58 

g 

.834189 

o  »7 

.863774 

1  97 

.970410 

.029584 

57 

4 

.834325 

2  °5 

.86365IJ 

1  97 

.970669 

'T^t 

56 

5 

6 

8 

.834460 
.834595 
.834730 
.834865 

2.25 
2.25 
2.25 

.863538  ! 
.863419 
.863301 
.863183 

1.98 
1.97  i 
1.97 

.9709^ 
.971175 
.971429  i 
.971682 

4.22 
4.23 
4.22 

.088885 

.028318 

55 
54 
53 

52 

9 

10 

.834999 
.835134 

2.25 
2.25 

.863064 
.862946 

1.97 
1.98 

.971935 
.972188 

4.22 
4.22 

.027812 

51 
50 

11 
12 
13 

9.835269 
.835403 
.835538 

2.23 
2.25 

9.862827 
.862709 
.862590 

1.97 
1.98 

1  QS   ' 

9.972441 
.972695 

.972948 

4.23 
4.88 

10.027559 
.027052 

49 

48 
47 

14 

.835672 

99 

.862471 

.973201 

.026799 

46 

15 
16 

.835807 
.835941 

2.23 

.862353 
.862234 

1.98  I 

.973454 
.973707  i 

4.22 

499 

.026293 

45 
44 

17 
18 

.836075 
.836209 

2.23 
290 

1  .862115 
.861996 

1.98 

.973960  i 
.974213 

4.22 

499 

.026040 
.025787 

43 
42 

19 

20 

.836343 
.836477 

2.23 
2  23 

.861877 

.861758 

1.98  ; 
2.00 

.974466 
.974720 

4.23 

4.22 

.025534 
.025280 

41 
40 

21 

9.836611 

2  ^3 

9.861638 

9.974973 

4  °2 

10.025027 

39 

22 

.836745 

!  .861519 

'   0 

.975226 

.024774 

38 

23 
24 
25 

26 
27 
28 
29 
30 

.836878 
.837012 
.837146 
.837279 
.837412 
.837546 
.837679 
.837812 

2.23 
2.23 
2.22 
2.22 
2.23 
2.22 
2.22 
2  22 

.861400 
.861280 
;  .861161 
.861041 
.860922 
.860802 
.860682 
i  .860562 

2.00 
1.98 
2.00 
1.98 
2.00 
2.00 
2.00 
2.00 

.975479 
.975732 
.9759a5 
.976238 
.976491 
.976744 
.976997 
.977250 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

.024521 
.084868 
.084015 

.023702 
.023509 
.088856 
.023003 
.022750 

37 
36 
35 
34 

as 

32 
31 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.837945 

.838078 
.838211 
.838344 
.838477 
.838610 
.838742 
.838875 
.839007 
.839140 

2.22 
2.22 
2.22 
2.22 
2.22 
2.20 
2.22 
2.20 
8.28 
2  k)0 

9.860442 
.860322 
.860202 
.860082 
.859962 
.859842 
.859721 
;   .859001 
i  .859480 
i  .859360 

2.00 
2.00 
2.00 
2.00 
2.00 
2.02 
2.00 
2.02 
2.00 
2  02 

9.977503 
.971756 
.978009 
.978262 
.978515 
.978768 
.979021 
.979274 
.979527 
.979780 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

10.022497 
.022244 
.021991 
.021738 
.U:214S5 
.021  •,>:!•,! 
.080979 

.080478 

.020220 

90 

28 
27 
26 
25 
24 
23 
22 
21 
80 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.839272 
.839404 

".839668 
.839800 
.839932 
.840064 
.840196 
.840328 
.840459 

9.840591 
.840722 
.840854 
.840985 
.841116 
.841247 
.841378 
.841509 
.841640 
9.841771 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.18 
2.20 

2.18 
2.20 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 

9.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
i  .858514 
.858393 
.858272 
.858151 

9.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857178 
.867066 
9.856934 

2.00 

2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.03 

2.02 
2.03 
2.02 
2.03 
2.02 
2.03 
2.03 
2.03 
2.03 

9.980033 
.980286 
.980538 
.980791 
.981044 
.981297 
.!  IS  1550 
.981803 
.982056 
.982309 

9.982562 
.982814 
.983067 
.983320 

.984079 

9.984837 

4  22 
4^20 
4.22 
i  4.22 
4.88 
4.88 
4.22 
4.22 
4.22 
4.22 

4.20 
4.22 

4.22 

4.22 

4.22 
4.20 
4.22 

10.019967 
.019714 

!  019809 

.018703 
.018450 
.018197 
.017944 
.017691 

10.017438 
.010088 

.016081 
!015668 

10.015163 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

y 

8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

1  Cosine. 

D.  1". 

Sine. 

i  D.  r. 

Cotang. 

D.r. 

Tang. 

1  ' 

AM* 

133° 


213 


44° 


TABLE   XII. — LOGARITHMIC    SINES, 


135° 


' 

Sine. 

D.  1'. 

Cosine. 

D.  I'. 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.841771 

9.856934 

2  no 

9.984837 

10.015163 

60 

1 

.841902 

2.18 

.850812 

.Uo 
o  no 

.985090 

499 

.014910 

59 

2 

.842033 

91^ 

.856690 

.985343 

,TC6 

.014657 

58 

3 
4 
5 

.842163 
.842294 
.842424 

2.18 
2.17 
21ft 

.856568 
.856446 
.856323 

2!  08 

2.05 

2  no 

.985596 
.985848 
.986101 

4.20 
4.22 

.014404 
.014152 
.013899 

57 
56 
55 

6 

7 
8 
9 

.842555 

.842685 
.842815 
.842946 

.  lo 
2.17 
2.17 

2.18 

.856201 
.856078 
.855956 
.855833 

.Uo 

2.05 
2.03 
2.05 

2  no 

.986354 
.986607 
.986860 
.987112 

4  '.22 
4.22 

4.20 

.013646 
.013393 
.013140 

.012888 

54 
53 
52 
51 

10 

.843076 

2.17 

.855711 

.Uo 

2.05 

.987365 

4.22 

.012635 

50 

11 
12 
13 
14 
15 
16 
17 

9.843206 
.843336 
.843466 
.843595 
.843725 
.843855 
.843984 

2.17 
2.17 
2.15 
2.17 
2.17 
2.15 

917 

9.855588 
.855465 
.855342 
.855219 
.855096 
.854973 
.854850 

2.05 
2.05 
2.05 
2.05 
2.05 
2.05 

9.987618 
.987871 
.988123 
.988376 
.988629 
.988882 
.989134 

4.22 
4.20 
4.22 
4.22 
4.22 
4.20 

10.012382 
.012129 
.011877 
.011624 
.011371 
.011118 
.010866 

49 
48 
47 
46 
45 
44 
43 

18 
19 

.844114 
.844243 

J6.lt 

2.15 

21  K 

.854727 
.854603 

2.05 
2.07 

.989387 
.389640 

4.22 
4.22 

.010613 
.010360 

42 
41 

20 

.844372 

.  JO 

2.17 

.854480 

2.05 
2.07 

.989893 

4^0 

.010107 

40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.844502 
.844631 
.844760 
.844889 
.845018 
.845147 
.845276 
.845405 
.845533 
.845662 

2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.13 
2.15 
2.13 

9.854356 
.854233 
.854109 
.853986 
.853862 
.853738 
.853614 
.85349^ 
.853366 
.853242 

2.05 
2.07 
2.05 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 

9.990145 
.990398 
.990651 
.990903 
.991156 
.991409 
.991662 
.991914 
.992167 
.992420 

4.22 
4  22 

4^20 

4  90 

4^22 

4  90 

4.  '20 
4.22 
4.22 
4.20 

10.009855 
.009602 
.009349 
.009097 
.008844 
.008591 
.008338 
.008086 
.007833 
.007580 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.845790 
.845919 
.846047 
.846175 
.846304 
.846432 
.846560 
.846688 
.846816 
.846944 

2.15 
2.13 
2.13 
2.15 
2.13 
2.13 
2.13 
2.13 
2.13 
2.12 

9.853118 
.852994 
.852869 
.852745 
.852620 
.852496 
.852371 
.852247 
.852122 
.851997 

2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.08 
2.08 

9.992672 
.992925 
.993178 
.993431 
.993683 
.993936 
.994189 
.994441 
.994694 
.994947 

4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 

10.007328 
.007075 
.006822 
.006569 
.006317 
.006064 
.005811 
.005559 
.005306 
.005053 

29 
28 
27 
26 
25 
24 
23 
oo 

21 
20 

41 

9.847071 

9  1Q 

9.851872 

9.995199 

10.004801 

19 

42 

.847199 

21  °. 

.851747 

2.08 

.995452 

4.22 

.004548 

18 

43 

44 
45 
46 
47 
48 
49 
50 

.847327 
.847454 

.847582 
.847709 
.847836 
.847964 
.848091 
.848218 

.  lu 

2.12 
2.13 
2.12 
2.12 
2.13 
2.12 
2.12 
2.12 

.851622 
.851497 
.851372 
.851246 
.851121 
.850998 
.850870 
.850745 

2.08 
2.08 
2.08 
2.10 
2.08 
2.08 
2.10 
2.08 
2.10 

.995705 
.995957 
.996210 
.996463 
.996715 
.996968 
.997221 
.997473 

4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 

.004295 
.004043 
.003790 
.003537 
.003285 
.003032 
.002779 
.002527 

17 
16 
15 
14 
13 
12 
11 
10 

51 

9.848345 

219 

9.850619 

9.997726 

49O 

10.002274 

9 

52 
53 

.848472 
.848599 

.  l~ 

2  12 

219 

.850493 
.850368 

2  .  10 

2.08 

.997979 
.998231 

.  4£ 
4.20 

.002021 
.001769 

8 

7 

54 
55 
56 
57 
58 
59 
60 

.848726 
.848852 
.848979 
.849106 
.849232 
.849359 
9.849485 

.  14 

2.10 
2.12 
2.12 
2.10 
2.12 
2.10 

.850242 
.850116 
.849990 
.849864 
.849738 
.849611 
9  84^485 

2.  10 
2.10 
2.10 
210  i 
2.10  i 
2.12  i 
2.10 

.998484 
.998737 
.998989 
.999242 
,999495 
.999747 
0.000000 

4.  22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 

.001516 
.001263 
.001011 
.000758 
.000505 
.000253 
10.000000 

6 
5 
4 
3 
o 

1 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1'  .  j  i  Cotang. 

D.  r. 

Tang. 

9 

134° 


220 


45' 


INDEX. 

(Names  of  animals  are  to  be  looked  for  under  their  class  name.) 

Amphibia,  variability 66 

Amphipoda,  see  Crustacea 67 

Ancestral  heredity 7g 

Annelida,  correlation 75 

,  variability ...  67 

Aphidae,  see  Hexapoda 66 

Area,  measurement  of 5 

Arithmetical  work,  precautions  in g 

Arithometer 7 

Assortative  mating 75 

Average 13,  17 

deviation. m 16 

Aves,  correlation ' 77 

,  variability 65 

of  eggs 65 

Bimodal  frequency  polygons 73 

Birds,  see  Aves. 

Brachiqpoda,  variability 67 

Brunsviga  calculator. 8 

Bryophyta,  variability 71 

Bryozoa,  correlation 77 

,  heredity 80 

,  variability 67 

Calculating  machines 7 

tables 7 

Caprifoliaceae,  variability 70 

Caryophyllacese,  variability 67 

Character  denned 1 

Chauvenet  's  criterion 12 

Class,  denned 

range 

Closeness  of  fit 2 

Coefficient  of  correlation.  . .. 4 

regression 47 

variability 16,  63 

Crelenterata,  see  Hydromedusa. 

Color,  measurement  of _'• 

Compositae,  correlation 'i 

,  variability 69,  70 

Comptometer ~ 

Coordinate  paper.     J* 

Cornaceae,  variability JJ 

Correlated  variability *j 

Counting,  methods  of ; 

Crabs f 

Criminals,  skull  index »• 

Critical  function z  \ 

Cruciferse,  variability 

223 


224  IKDEX, 

PAGE 

Crustacea,  Amphipoda,  variability 67 

,  correlation 76 

,  Daphnia,  correlation 77 

,  heredity 79,  80 

,  Eupagurus,  correlation 77 

,  local  races 84 

,  variability 63,  66 

Decimal  places,  number  to  employ 8 

Dipsacse,  variability 70 

Discontinuous  variates 1 

Dissymmetrical  animals,  bilateral  correlation  of 76 

Dissymmetry 82 

index 60 

Dominating  characters 58 

Echinodermata,  correlation 76 

,  variability 68 

Environment,  direct  effect  of .   83 

Fertility,  heredity  of 82 

Fishes,  see  Pisces. 

Frequency  polygon 62 

Fruit,  variability  of 71 

Galton  's  difference  problem 27 

Gastropoda,  correlation 77 

,  variability 67 

Geometric  mean -. 15 

Graduated  variates 1 

Heredity 55,  78 

,  ancestral 78 

Hexapoda,  correlation 77 

,  variability 66 

Homo   correlation 73 

eye-color,  heredity  of 79 

fertility,  heredity  of 79,  80 

inheritance 79 

head  index,  heredity  of 79 

mental  characters,  heredity  of 80 

skeletal,  correlation 74 

skull,  variability  of 64 

stature,  correlation 79 

weight,  variability 63 

variability 64 

(See  also  Naquada  race) 64,  65,  74 

Homotyposis 81 

Hydromedusae,  variability 68 

Index  of  abmodality 23 

dissymmetry 6 

divergence 40 

isolation 41 

variability 15,  17 

Individual 1 

variation 1 

vs.  specific  variation 63 

Integral  variate 1 

Lamellibranchiata,  correlation 76 

,  local  races 84 

,  variability 68,  71 

Leaves,  variability 71 

Leguminosse,  variability 70 

Lepidoptera,  variability 66 

Loaded  ordinates,  method  of 12 

Local  races 83 

Longevity,  inheritance  of 79 

Mammalia,  correlation. 76 

,  variability 65 

Mean 13 

Median.  . .  . 14 


INDEX.  225 


Mendelism  ...........................................  &G£ 

Mid-departure  ................. 

Mode  .............................  :::::::::;: 

Multimodal  polygons  ..............................                   '39  73 

Multiple  organ  ...............................  '  ' 

Mutations  ................................  ........  .63 

Myriapoda,  correlation  ...........................  . 

,  variability  ........................  ........              !  !  66 

Naquada  race,  skeletal  variability  ....................           64,  65,  74 

Normal  curve  of  frequency  ......................... 

Number  of  variates  to  employ  .................  ......... 

Orchidacese,  variability  ..............................  '  [             \\  71 

Organ  variation  .................................  .  .  . 

Papaveracese,  variability  .............................  70 

Partial  variation  ..................................  ........  1 

Person  ...................................................    '  i 

Pisces  correlation  ....................................  .....  76,  77 

local  races   .  .  .........................................  83 

variability  ........................................      .  .  66 

Plants   correlation  ...........................................  7g 

homotyposis  ..........................................  81 

variability  .......................................  .  .„  .  69 

Prepotency  ..................................................  78 

Primulacese,  variability  ......................................  70 

Probable  departure  ...........................................  16 

difference  ...........................................  15 

error.  ..............................................  14 

in  uniparental  heredity  ...........................  55 

of  coefficient  of  correlation  .......................  44 

of  variability  ........  *  ..............  16 

of  mean  .......................................  15 

of  median  ......................................  15 

of  standard  deviation  ............................  16 

Probability  of  normality  of  a  given  distribution  ..................  24 

Protista,  correlation  ..........................................  77 

,  variability  ..........................................  69 

Range  of  variability  ..........................................  25 

Ranunculaceae,  variability  .....................................  69 

Recessive  characters  ..............................  ............  58 

Rectangles,  method  of,  in  platting  frequency  distributions  .........  11 

Rejection  of  extreme  variates  ..................................  12 

Relative  variability  of  the  sexes  ................................  63 

Rosacese,  variability  ..........................................  70 

Sapidacese,  variability.  .  .  .....................................  70 

Scrophulariaceae,  variability  ........  .  ..........................  71 

Selection  .......  ....  .........................................  82 

Sex,  relative  variability  .......................................  63 

Sedation  ....................................................  10 

Skewness  .................  .............................  30,  71,  72 

Skull,  see  Homo. 

Spurious  correlation  .....  .  ....................................  54 

Standard  deviation  ...........................................  16 

Stature,  see  Homo. 

Symmetry  in  frequency  distribution  ............................  1 

Telegony  ....................................................  82 

Types  of  frequencydistribution  ..........................  19,  71,  72 

Variability  .........................................  15,  17,  62-71 

Variant  ..........................................  ........... 

Variate  .....................................................  1 

Weight,  variability,  see  Homo. 


UNIVERSITY   OF    CALIFORNIA 
BRANCH    OF    THE    COLLEGE    OF    AGRICULTURE 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


6  1937 


FEB     5  1946 
£  8  1947 

194?' 


38395 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


